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The summary table makes it easier to identify patterns in the data. Recall that the question of interest is whether children in the peanut consumption group are more or less likely to develop peanut allergies than those in the peanut avoidance group. In the avoidance group, the proportion of children failing the OFC is... | {
"Header 1": "**1.1 Case study: preventing peanut allergies**",
"Header 3": "1.1. CASE STUDY 13",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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In evolutionary biology, parental investment refers to the amount of time, energy, or other resources devoted towards raising offspring. This section introduces the frog dataset, which originates from a 2013 study about maternal investment in a frog species.[4](#page-13-1) Reproduction is a costly process for female fr... | {
"Header 1": "**1.2 Data basics**",
"Header 3": "**1.2.1 Observations, variables, and data matrices**",
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The Functional polymorphisms Associated with human Muscle Size and Strength study (FA-MuSS) measured a variety of demographic, phenotypic, and genetic characteristics for about 1,300 participants.[7](#page-14-0) Data from the study have been used in a number of subsequent studies,[8](#page-14-1) such as one examining t... | {
"Header 1": "**1.2 Data basics**",
"Header 3": "**1.2.2 Types of variables**",
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Many studies are motivated by a researcher examining how two or more variables are related. For example, do the values of one variable increase as the values of another decrease? Do the values of one variable tend to differ by the levels of another variable?
One study used the famuss data to investigate whether ACTN3... | {
"Header 1": "**1.2 Data basics**",
"Header 3": "**1.2.3 Relationships between variables**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Consider the following research questions:
- 1. Do bluefin tuna from the Atlantic Ocean have particularly high levels of mercury, such that they are unsafe for human consumption?
- 2. For infants predisposed to developing a peanut allergy, is there evidence that introducing peanut products early in life is an effecti... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.1 Populations and samples**",
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Anecdotal evidence typically refers to unusual observations that are easily recalled because of their striking characteristics. Physicians may be more likely to remember the characteristics of a single patient with an unusually good response to a drug instead of the many patients who did not respond. The dangers of dra... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.2 Anecdotal evidence**",
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Sampling from a population, when done correctly, provides reliable information about the characteristics of a large population. The US Centers for Disease Control (US CDC) conducts several surveys to obtain information about the US population, including the Behavior Risk Factor Surveillance System (BRFSS).[19](#page-19... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.3 Sampling from a population**",
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Almost all statistical methods are based on the notion of implied randomness. If data are not sampled from a population at random, these statistical methods – calculating estimates and errors associated with estimates – are not reliable. Four random sampling methods are discussed in this section: simple, stratified, cl... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.4 Sampling methods**",
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The two primary types of study designs used to collect data are experiments and observational studies.
In an experiment, researchers directly influence how data arise, such as by assigning groups of individuals to different treatments and assessing how the outcome varies across treatment groups. The LEAP study is an ... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.5 Introducing experiments and observational studies**",
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Experimental design is based on three principles: control, randomization, and replication.
- Control. When selecting participants for a study, researchers work to control for extraneous variables and choose a sample of participants that is representative of the population of interest. For example, participation in a ... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.6 Experiments**",
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In observational studies, researchers simply observe selected potential explanatory and response variables. Participants who differ in important explanatory variables may also differ in other ways that influence response; as a result, it is not advisable to make causal conclusions about the relationship between explana... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**1.3.7 Observational studies**",
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As stated in Example [1.4,](#page-16-2) female body size (body.size) in the parental investment study is neither an explanatory nor a response variable. Previous research has shown that larger females tend to produce larger eggs and egg clutches; however, large body size can be costly at high altitudes. Discuss a possi... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "**GUIDED PRACTICE 1.9**",
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Observational studies may reveal interesting patterns or associations that can be further investigated with follow-up experiments. Several observational studies based on dietary data from different countries showed a strong association between dietary fat and breast cancer in women. These observations led to the launch... | {
"Header 1": "**1.3 Data collection principles**",
"Header 3": "1.3. DATA COLLECTION PRINCIPLES 29",
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The sample mean of a numerical variable is the sum of the values of all observations divided by the number of observations:
$$\overline{x} = \frac{x_1 + x_2 + \dots + x_n}{n},$$
(1.10)
where *x*1*, x*2*,..., x<sup>n</sup>* represent the *n* observed values.
The median is another measure of center; it is the middl... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**MEAN**",
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The spread of a distribution refers to how similar or varied the values in the distribution are to each other; i.e., whether the values are tightly clustered or spread over a wide range.
The standard deviation for a set of data describes the typical distance between an observation and the mean. The distance of a sing... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "1.4.2 Measures of spread: standard deviation and interguartile range",
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The sample standard deviation of a numerical variable is computed as the square root of the variance, which is the sum of squared deviations divided by the number of observations minus 1.
$$S = \sqrt{\frac{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 + \dots + (x_n - \overline{x})^2}{n - 1}},$$
(1.11)
where $x_... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**STANDARD DEVIATION**",
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Figure [1.15](#page-31-1) shows the values of clutch.volume as points on a single axis. There are a few values that seem extreme relative to the other observations: the four largest values, which appear distinct from the rest of the distribution. How do these extreme values affect the value of the numerical summaries? ... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**1.4.3 Robust estimates**",
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Graphs show important features of a distribution that are not evident from numerical summaries, such as asymmetry or extreme values. While dot plots show the exact value of each observation, histograms and boxplots graphically summarize distributions.
In a histogram, observations are grouped into bins and plotted as ... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**1.4.4 Visualizing distributions of data: histograms and boxplots**",
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When working with strongly skewed data, it can be useful to apply a transformation, and rescale the data using a function. A natural log transformation is commonly used to clarify the features of a variable when there are many values clustered near zero and all observations are positive.
 ... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**1.4.5 Transforming data**",
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On the log-transformed scale, mean log income is 8.50, with standard deviation 1.54. Apply the empirical rule to describe the distribution of average yearly per capita income among the 165 countries.
According to the empirical rule, the middle 70% of the data are within one standard deviation of the mean, in the rang... | {
"Header 1": "**1.4 Numerical data**",
"Header 3": "**EXAMPLE 1.13**",
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This section introduces tables and plots for summarizing categorical data, using the famuss dataset introduced in Section [1.2.2.](#page-14-6)
A table for a single variable is called a frequency table. Figure [1.23](#page-36-1) is a frequency table for the actn3.r577x variable, showing the distribution of genotype at... | {
"Header 1": "**1.5 Categorical data**",
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In the frog parental investment study, researchers used clutch volume as a primary variable of interest rather than egg size because clutch volume represents both the eggs and the protective gelatinous matrix surrounding the eggs. The larger the clutch volume, the higher the energy required to produce it; thus, higher ... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "Scatterplots",
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Body mass index (BMI) is a measure of weight commonly used by health agencies to assess whether someone is overweight, and is calculated from height and weight.<sup>39</sup> Describe the relationships shown in Figure 1.27. Why is it helpful to use BMI as a measure of obesity, rather than weight?
Figure 1.27(a) shows ... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "EXAMPLE 1.14",
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Figure 1.28 is a scatterplot of life expectancy versus annual per capita income for 165 countries in 2011. Life expectancy is measured as the expected lifespan for children born in 2011 and income is adjusted for purchasing power in a country. Describe the relationship between life expectancy and annual per capita inco... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "EXAMPLE 1.15",
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**Correlation** is a numerical summary statistic that measures the strength of a linear relationship between two variables. It is denoted by *r*, the **correlation coefficient**, which takes on values between -1 and 1.
If the paired values of two variables lie exactly on a line, $r = \pm 1$ ; the closer the correlat... | {
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"Header 3": "Correlation",
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Calculate the correlation coefficient of *x* and *y*, plotted in Figure [1.30.](#page-41-0)
Calculate the mean and standard deviation for *x* and *y*: *x* = 2, *y* = 3, *s<sup>x</sup>* = 1, and *s<sup>y</sup>* = 2*.*65.
$$r = \frac{1}{n-1} \sum_{i=1}^{n} \left( \frac{x_i - \overline{x}}{s_x} \right) \left( \frac{y_... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "**EXAMPLE 1.17**",
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A **contingency table** summarizes data for two categorical variables, with each value in the table representing the number of times a particular combination of outcomes occurs.<sup>42</sup> Figure 1.32 summarizes the relationship between race and genotype in the famuss data.
The **row totals** provide the total coun... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "**Contingency tables**",
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A segmented bar plot is a way of visualizing the information from a contingency table. Figure [1.35](#page-44-0) graphically displays the data from Figure [1.32;](#page-42-2) each bar represents a level of actn3.r577x and is divided by the levels of race. Figure [1.35\(b\)](#page-44-1) uses the row proportions to creat... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "Segmented bar plots",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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The results from medical studies are often presented in two-by-two tables (2 × 2 tables), contingency tables for categorical variables that have two levels. One of the variables defines two groups of participants, while the other represents the two possible outcomes. Figure [1.37](#page-45-0) shows a hypothetical two-b... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "Two-by-two tables: relative risk",
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Suppose another study to examine the association between smoking and cardiovascular disease is conducted, but researchers use a different study design than described in Example [1.22.](#page-46-0) For the new study, 90 individuals with CVD and 110 individuals without CVD are recruited. 40 of the individuals with CVD ar... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "**EXAMPLE 1.24**",
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Methods for comparing numerical data across groups are based on the approaches introduced in Section [1.4.](#page-29-0) Side-by-side boxplots and hollow histograms are useful for directly comparing how the distribution of a numerical variable differs by category.
Recall the question introduced in Section [1.2.3:](#pa... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "**1.6.3 A numerical variable and a categorical variable**",
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Using Figure [1.40,](#page-48-1) assess how maternal investment varies with altitude.[48](#page-47-1)
<sup>46</sup>Tea drinking habits and oesophageal cancer in a high risk area in northern Iran: population based casecontrol study, Islami F, et al., BMJ (2009), doi <10.1136/bmj.b929>
<sup>47</sup>The proportions ca... | {
"Header 1": "**1.6 Relationships between two variables**",
"Header 3": "**GUIDED PRACTICE 1.26**",
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In the United States, individuals with developmental disabilities typically receive services and support from state governments. The State of California allocates funds to developmentallydisabled residents through the California Department of Developmental Services (DDS); individuals receiving DDS funds are referred to... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**1.7.1 Case study: discrimination in developmental disability support**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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To begin understanding a dataset, start by examining the distributions of single variables using numerical and graphical summaries. This process is essential for developing a sense of context; in this case, examining variables individually addresses questions such as "What is the range of annual expenditures?", "Do con... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "Distributions of single variables",
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After examining variables individually, explore how variables are related to each other. While there exist methods for summarizing more than two variables simultaneously, focusing on two variables at a time can be surprisingly effective for making sense of a dataset. It is useful to begin by investigating the relations... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "Relationships between two variables",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Figure [1.50](#page-54-0) compares the distribution of expenditures between Hispanic and White non-Hispanic consumers. Most Hispanic consumers receive between about \$0 to \$20,000 from the California DDS; individuals receiving amounts higher than this are upper outliers. However, for White non-Hispanic consumers, medi... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "1.7. EXPLORATORY DATA ANALYSIS 55",
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Instead, it is the difference in age distributions of the two populations that is driving the observed discrepancy in expenditures. The overall average of expenditures for the Hispanic consumers is lower because the population of Hispanic consumers is relatively young compared to the population of White non-Hispanic co... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "1.7. EXPLORATORY DATA ANALYSIS 57",
"token_count": 231,
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Using the proportions in Figure [1.52](#page-55-0) and the average expenditures for each cohort in Figure [1.53,](#page-55-1) calculate the overall weighted average expenditures for Hispanics and for White non-Hispanics.[52](#page-56-0)
For Hispanics:
1*,*393(*.*12) + 2*,*312(*.*24) + 3*,*955(*.*27) + 9*,*960(*.*21... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**EXAMPLE 1.29**",
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The genetic code stored in DNA contains the necessary information for producing the proteins that ultimately determine an organism's observable traits (phenotype). Although nearly every cell in an organism contains the same genes, cells may exhibit different patterns of gene expression. Not only can genes be switched o... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**1.7.2 Case study: molecular cancer classification**",
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Microarray technology is based on hybridization, a basic property of nucleic acids in which complementary nucleotide sequences specifically bind together. Each microarray consists of a glass or silicon slide dotted with a grid of short (25-40 base pairs long), single-stranded DNA fragments, known as probes. The probes ... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "DNA microarrays",
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Accurate cancer classification is critical for determining an appropriate course of therapy. The chemotherapy regimens for acute leukemias differs based on whether the leukemia affects bloodforming cells (acute myeloid leukemia, AML) or white blood cells (acute lymphoblastic leukemia, ALL). At the time of the Golub stu... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "Golub leukemia study",
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On average, which genes are more highly expressed in AML patients? Which genes are more highly expressed in ALL patients?
For each gene, compare the mean expression value among ALL patients to the mean among AML patients. For example, the difference in mean expression levels for Gene A is
*xAML* − *xALL* = 45186 − ... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**EXAMPLE 1.30**",
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Consider the expression data for the patient in the first row of Figure [1.63.](#page-64-0) For each gene, identify whether the expression level is more AML-like or more ALL-like.
For the gene represented by the M19507\_at probe, the patient has a recorded expression level of 4,481, which is closer to the ALL mean of... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**EXAMPLE 1.31**",
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Use the information in Figures [1.63](#page-64-0) and [1.64](#page-64-1) to calculate the magnitude of the deviations *v*<sup>1</sup> and *v*<sup>10</sup> for the first patient.
For the gene represented by the M19507\_at probe, the magnitude of the deviation is *v*<sup>1</sup> = |4*,*481 − 10*,*232| = 5*,*751.
For ... | {
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"Header 3": "**EXAMPLE 1.32**",
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Make a prediction for the leukemia status of Patient 10.[61](#page-65-1)
Figure [1.66](#page-65-2) shows the comparison between actual leukemia status and predicted leukemia status based on the described prediction strategy. The prediction matches patient leukemia status for all patients.
| | Actual | Prediction... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**GUIDED PRACTICE 1.34**",
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In contrast to hybridization-based approaches, RNA sequencing (RNA-Seq) allows for the entire transcriptome to be surveyed in a high-throughput, quantitative manner.[62](#page-66-0) Microarrays require gene-specific probes, which limits microarray experiments to detecting transcripts that correspond to known gene seque... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "**1.7.3 Case study: cold-responsive genes in the plant** *Arabidopsis arenosa*",
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The first step in an RNA-Seq experiment is to prepare cDNA sequence libraries for each RNA sample being sequenced. RNA is converted into cDNA and sheared into short fragments; sequencing adapters and barcodes are added to each fragment that initiate the sequencing reaction and identify sequences that originate from dif... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "RNA sequencing (RNA-Seq)",
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*Arabidopsis arenosa* populations exist in different habitats, and exhibit a range of differences in flowering time, cold sensitivity, and perenniality. Sensitivity to cold is an important trait for perennials, plants that live longer than one year. It is common for perennials to require a period of prolonged cold in o... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "Cold-responsive genes in *A. arenosa*",
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| | Gene Name | KA NV 1 | KA NV 2 | KA NV 3 | KA V 1 | KA V 2 | KA V 3 |
|---|-----------|---------|---------|---------|---------|---------|---------|
| 1 | PUX4 | 288.20 | 322.55 | 305.35 | 1429.29 | 1408.25 | 1487.08 |
| 2 | TZP | 79.36 | 93.34 | 73.44 | 1203.40 | 1230.49 | 1214.03 |
| 3 | GA... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "1.7. EXPLORATORY DATA ANALYSIS 69",
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Figure [1.72](#page-70-0) shows the log2-transformed expression ratios as a side-by-side boxplot.[66](#page-69-5)
| | Gene Name | TBG | KA | | Gene Name | TBG | KA |
|---|-----------|------|-------|---|-----------|-------|------|
| 1 | PUX4 | 2.06 | 4.72 | 1 | PUX4 | 1.04 | 2.24 |
| 2 | TZP ... | {
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There are many ways to numerically and graphically summarize data that are not explicitly introduced in this chapter. Presentation-style graphics in published manuscripts can be especially complex, and may feature techniques specific to a certain field as well as novel approaches designed to highlight particular featur... | {
"Header 1": "**1.7 Exploratory data analysis**",
"Header 3": "Advanced data visualization",
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Introductory treatments of statistics often emphasize the value of formal methods of probability and inference, topics which are covered in the remaining chapters of this text. However, numerical and graphical summaries are essential for understanding the features of a dataset and should be applied before the process o... | {
"Header 1": "**1.8 Notes**",
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**[1.3](#page-434-3) Air pollution and birth outcomes, study components.** Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution levels were measured by air quality monitoring stations. Specifically, levels of carbon monox... | {
"Header 1": "1.9 Exercises",
"Header 3": "**1.9.2 Data basics**",
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**1.10** Cheaters, scope of inference. Exercise 1.5 introduces a study where researchers studying the relationship between honesty, age, and self-control conducted an experiment on 160 children between the ages of 5 and 15. The researchers asked each child to toss a fair coin in private and to record the outcome (white... | {
"Header 1": "1.9 Exercises",
"Header 3": "1.9.3 Data collection principles",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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**[1.13](#page-434-13) Buteyko method, scope of inference.** Exercise [1.4](#page-75-5) introduces a study on using the Buteyko shallow breathing technique to reduce asthma symptoms and improve quality of life. As part of this study 600 asthma patients aged 18-69 who relied on medication for asthma treatment were recru... | {
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"Header 3": "1.9. EXERCISES 79",
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**[1.21](#page-435-6) Flawed reasoning.** Identify the flaw(s) in reasoning in the following scenarios. Explain what the individuals in the study should have done differently if they wanted to make such conclusions.
- (a) Students at an elementary school are given a questionnaire that they are asked to return after t... | {
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"Header 3": "1.9. EXERCISES 81",
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**[1.38](#page-436-11) Smoking and stenosis.** Researchers collected data from an observational study to investigate the association between smoking status and the presence of aortic stenosis, a narrowing of the aorta that impedes blood flow to the body.
| | | Smoking Status | | |
... | {
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"Header 3": "1.9. EXERCISES 87",
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- **[2.1](#page-89-0) [Defining probability](#page-89-0)**
- **[2.2](#page-105-0) [Conditional probability](#page-105-0)**
- **[2.3](#page-116-0) [Extended example](#page-116-0)**
- **[2.4](#page-125-0) [Notes](#page-125-0)**
- **[2.5](#page-126-0) [Exercises](#page-126-0)**
What are the chances that a woman with an ... | {
"Header 1": "**Probability**",
"token_count": 438,
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Consider rolling two fair dice. What is the chance of getting two 1s?
If 1*/*6 *th* of the time the first die is a 1 and 1*/*6 *th* of *those* times the second die is also a 1, then the chance that both dice are 1 is (1*/*6)(1*/*6) or 1*/*36.
Probability can also be used to model less artificial contexts, such as t... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.3**",
"token_count": 278,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Suppose that both members of a couple are CF carriers. What is the probability that a child of this couple will be affected by CF? Assume that a parent has an equal chance of passing either gene copy (i.e., allele) to a child.
*Solution 1: Enumerate all of the possible outcomes and exploit the fact that the outcomes ... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.4**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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The probability of an outcome is the proportion of times the outcome would occur if the random phenomenon could be observed an infinite number of times.
This definition of probability can be illustrated by simulation. Suppose a die is rolled many times. Let *p*ˆ*<sup>n</sup>* be the proportion of outcomes that are 1 ... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**PROBABILITY**",
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As more observations are collected, the proportion *p*ˆ*<sup>n</sup>* of occurrences with a particular outcome converges to the probability *p* of that outcome.
Probability is defined as a proportion, and it always takes values between 0 and 1 (inclusively). It may also be expressed as a percentage between 0% and 100... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**LAW OF LARGE NUMBERS**",
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Consider the CF example. Is the event that two carriers of CF have a child that is also a carrier represented by mutually exclusive outcomes? Calculate the probability of this event.<sup>4</sup>
Probability problems often deal with *sets* or *collections* of outcomes. Let *A* represent the event in which a die roll r... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.7**",
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(G)
(a) Using Figure 2.3 as a reference, which outcomes are represented by event D? (b) Are events B and D disjoint? (c) Are events A and D disjoint?<sup>6</sup>
G
<sup>&</sup>lt;sup>4</sup>Yes, there are two mutually exclusive outcomes for which a child of two carriers can also be a carrier - a child can either ... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.9**",
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(a) What is the probability that a randomly selected card is a diamond? (b) What is the probability that a randomly selected card is a face card?<sup>8</sup>
<sup>&</sup>lt;sup>7</sup>Since *B* and *D* are disjoint events, use the Addition Rule: $P(B \text{ or } D) = P(B) + P(D) = \frac{1}{3} + \frac{1}{3} = \frac{2... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.11**",
"token_count": 531,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Human immunodeficiency virus (HIV) and tuberculosis (TB) affect substantial proportions of the population in certain areas of the developing world. Individuals sometimes are co-infected (i.e., have both diseases). Children of HIV-infected mothers may have HIV and TB can spread from one family member to another. In a mo... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.15**",
"token_count": 333,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
A probability distribution is a list of all possible outcomes and their associated probabilities that satisfies three rules:
- 1. The outcomes listed must be disjoint.
- 2. Each probability must be between 0 and 1.
- 3. The probabilities must total to 1.
Figure 2.7: The probability di... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**RULES FOR A PROBABILITY DISTRIBUTION**",
"token_count": 471,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Probability distributions for events that take on a finite number of possible outcomes, such as the sum of two dice rolls, are referred to as discrete probability distributions.
Consider how the probability distribution for adult heights in the US might best be represented. Unlike the sum of two dice rolls, height ca... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "Continuous probability distributions",
"token_count": 240,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Estimate the probability that a randomly selected adult from the US population has height between 180 and 185 centimeters. In Figure [2.10\(a\),](#page-98-0) the two bins between 180 and 185 centimeters have counts of 195,307 and 156,239 people.
Find the proportion of the histogram's area that falls in the range 180 ... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.16**",
"token_count": 341,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
S Sample space
A<sup>c</sup> Complement of outcome A Rolling a die produces a value in the set $\{1, 2, 3, 4, 5, 6\}$ . This set of all possible outcomes is called the **sample space** (*S*) for rolling a die.
Let $D = \{2, 3\}$ represent the event that the outcome of a die roll is 2 or 3. The **complement** of ... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "2.1.6 Complement of an event",
"token_count": 272,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
(G)
Events $A = \{1, 2\}$ and $B = \{4, 6\}$ are shown in Figure 2.3 on page 94. (a) Write out what $A^c$ and $B^c$ represent. (b) Compute $P(A^c)$ and $P(B^c)$ . (c) Compute $P(A) + P(A^c)$ and $P(B) + P(B^c)$ .<sup>16</sup>
A complement of an event A is constructed to have two very important propert... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.20**",
"token_count": 577,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Just as variables and observations can be independent, random phenomena can also be independent. Two processes are **independent** if knowing the outcome of one provides no information about the outcome of the other. For instance, flipping a coin and rolling a die are two independent processes – knowing that the coin l... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "2.1.7 Independence",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Mandatory drug testing. Mandatory drug testing in the workplace is common practice for certain professions, such as air traffic controllers and transportation workers. A false positive in a drug screening test occurs when the test incorrectly indicates that a screened person is an illegal drug user. Suppose a mandatory... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.30**",
"token_count": 345,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Because of the high likelihood of at least one false positive in company wide drug screening programs, an individual with a positive test is almost always re-tested with a different screening test: one that is more expensive than the first, but has a lower false positive probability. Suppose the second test has a false... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**GUIDED PRACTICE 2.31**",
"token_count": 303,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
ABO blood groups. There are four different common blood types (A, B, AB, and O), which are determined by the presence of certain antigens located on cell surfaces. Antigens are substances used by the immune system to recognize self versus non-self; if the immune system encounters antigens not normally found on the body... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.32**",
"token_count": 614,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Is the event of drawing a heart from a deck of cards independent of drawing an ace?
The probability the card is a heart is 1*/*4 (13*/*52 = 1*/*4) and the probability that it is an ace is 1*/*13 (4*/*52 = 1*/*13). The probability that the card is the ace of hearts (A♥) is 1*/*52. Check whether Equation [2.29](#page-1... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.33**",
"token_count": 202,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
In the general population, about 15% of adults between 25 and 40 years of age are hypertensive. Suppose that among males of this age, hypertension occurs about 18% of the time. Is hypertension independent of sex?
Assume that the population is 50% male, 50% female; it is given in the problem that hypertension occurs a... | {
"Header 1": "**2.1 Defining probability**",
"Header 3": "**EXAMPLE 2.34**",
"token_count": 223,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
While it is difficult to obtain precise estimates, the US CDC estimated that in 2012, approximately 29.1 million Americans had type 2 diabetes – about 9.3% of the population.[23](#page-105-1) A health care practitioner seeing a new patient would expect a 9.3% chance that the patient might have diabetes.
However, this... | {
"Header 1": "**2.2 Conditional probability**",
"token_count": 253,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Figures [2.14](#page-105-2) and [2.15](#page-106-0) provide additional information about the relationship between diabetes prevalence and age.[24](#page-105-3) Figure [2.14](#page-105-2) is a contingency table for the entire US population in 2012; the values in the table are in thousands (to make the table more readabl... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**2.2.1 Marginal and joint probabilities**",
"token_count": 634,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
What fraction of the US population are 45 to 64 years of age and have diabetes? What fraction of the population age 45 to 64 have diabetes?[25](#page-106-1)
The entries in Figure [2.15](#page-106-0) show the proportions of the population in each of the eight categories defined by diabetes status and age, obtained by ... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**GUIDED PRACTICE 2.35**",
"token_count": 617,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
What is the interpretation of the value 0.907 in the last row of the table? And of the value 0.097 directly above it?[27](#page-106-3)
<sup>25</sup>The first value is given by the intersection of "45 - 64 years of age" and "diabetes", divided by the total population number: 13*,*400*,*000*/*313*,*320*,*000 = 0*.*043.... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**GUIDED PRACTICE 2.36**",
"token_count": 304,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The probability that a randomly selected individual from the US has diabetes is 0.093, the sum of the first column in Figure [2.15.](#page-106-0) How does that probability change if it is known that the individual's age is 64 or greater?
The conditional probability can be calculated from Figure [2.14,](#page-105-2) w... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**2.2.2 Defining conditional probability**",
"token_count": 513,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Calculate the probability a randomly selected person is older than 20 years of age, given that the person has diabetes. $^{30}$
G
**G**
<sup>&</sup>lt;sup>29</sup> Again, let *A* be the event a person has diabetes, and *B* the event that their age is between 45 and 64. Find P(B|A). $P(B|A) = \frac{P(A \text{ an... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**GUIDED PRACTICE 2.40**",
"token_count": 236,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Using the Multiplication Rule for independent events allows for a mathematical illustration of why the condition information has no influence in Example [2.42:](#page-109-0)
$$P(Y = 1 | X = 1) = \frac{P(Y = 1 \text{ and } X = 1)}{P(X = 1)}$$
$$= \frac{P(Y = 1)P(X = 1)}{P(X = 1)}$$
$$= P(Y = 1).$$
This is a specific... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "2.2. CONDITIONAL PROBABILITY 111",
"token_count": 220,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
This chapter began with a straightforward question – what are the chances that a woman with an abnormal (i.e., positive) mammogram has breast cancer? For a clinician, this question can be rephrased as the conditional probability that a woman has breast cancer, given that her mammogram is abnormal. This conditional prob... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**2.2.5 Bayes' Theorem**",
"token_count": 414,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Figure 2.16: A tree diagram for breast cancer screening.
A tree diagram is a tool to organize outcomes and probabilities around the structure of data, and is especially useful when two or more processes occur in a sequence, with each process conditioned on its predecessors.
In Figure ... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "Example [2.44](#page-111-1) Solution 1. Tree Diagram.",
"token_count": 1105,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The process used to solve the problem via the tree diagram can be condensed into a single algebraic expression by substituting the original probability expressions into the numerator and denominator:
$$P(\text{has BC} \mid \text{mammogram}^+) = \frac{P(\text{has BC and mammogram}^+)}{P(\text{mammogram}^+)}$$
$$= \fra... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "Example 2.44 Solution 2. Bayes' Rule.",
"token_count": 408,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Consider the following conditional probability for variable 1 and variable 2:
P(outcome $A_1$ of variable 1 | outcome B of variable 2).
Bayes' Theorem states that this conditional probability can be identified as the following fraction:
$$\frac{P(B|A_1)P(A_1)}{P(B|A_1)P(A_1) + P(B|A_2)P(A_2) + \dots + P(B|A_k)P... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**BAYES' THEOREM**",
"token_count": 331,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The positive predictive value (PPV) of a diagnostic test can be calculated by constructing a two-way contingency table for a large, hypothetical population and calculating conditional probabilities by conditioning on rows or columns. Using a large enough hypothetical population results in an empirical estimate of PPV t... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "Example [2.44](#page-111-1) Solution 3. Contingency Table.",
"token_count": 1111,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Some congenital disorders are caused by errors that occur during cell division, resulting in the presence of additional chromosome copies. Trisomy 21 occurs in approximately 1 out of 800 births. Cell-free fetal DNA (cfDNA) testing is one commonly used way to screen fetuses for trisomy 21. The test sensitivity is 0.98 a... | {
"Header 1": "**2.2 Conditional probability**",
"Header 3": "**GUIDED PRACTICE 2.47**",
"token_count": 337,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The gene that controls white coat color in cats, *KIT* , is known to be responsible for multiple phenotypes such as deafness and blue eye color. A dominant allele *W* at one location in the gene has complete penetrance for white coat color; all cats with the *W* allele have white coats. There is incomplete penetrance f... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "Problem statement",
"token_count": 400,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The following information has been given in the problem. Re-write the information using the notation defined earlier.
Suppose that 30% of white cats have one blue eye, while 10% of white cats have two blue eyes. About 73% of white cats with two blue eyes are deaf and 40% of white cats with one blue eye are deaf. Only... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.48**",
"token_count": 879,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Using the definition of conditional probability, solve for *P* (*B*2|*D*).
$$P(B_2|D) = \frac{P(D \text{ and } B_2)}{P(D)} = \frac{P(D|B_2)P(B_2)}{P(D)} = \frac{(0.73)(0.10)}{0.307} = 0.238.$$
The probability that a white cat has two blue eyes, given that it is deaf, is 0.238.
It is also possible to think of this... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.49**",
"token_count": 290,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The following information has been given in the problem. Re-write the information using the notation defined earlier.
Suppose that deaf, white cats have an increased chance of being blind, but that the prevalence of blindness differs according to eye color. While deaf, white cats with two blue eyes or two non-blue ey... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.50**",
"token_count": 334,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Expand the previous expression using the general multiplication rule, P(A and B) = P(A|B)P(B).
The general multiplication rule may seem difficult to apply when conditioning is present, but the principle remains the same. Think of the conditioning as a way to restrict the sample space; in this context, conditioning on... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.51**",
"token_count": 866,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Calculate the prevalence of blindness among white cats, *P* (*L*).
$$P(L) = P(L \text{ and } D) + P(L \text{ and } D^{C})$$
= $P(L|D)P(D) + P(L|D^{C})P(D^{C})$
= $(0.278)(0.307) + (0.10)(1 - 0.307)$
= $0.155.$
*P* (*D*) was calculated in part a), while *P* (*L*|*D*) was calculated in part c, i. The conditionin... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.53**",
"token_count": 420,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Draw a tree diagram to organize the events involved in this problem. Identify the branches that represent the possible paths for a white cat to both have two blue eyes and be blind.
When drawing a tree diagram, remember that each branch is conditioned on the previous branches. While there are various possible trees, ... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.55**",
"token_count": 380,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Expand Equation [2.54](#page-120-0) according to the tree shown in Figure [2.19\(a\),](#page-121-1) and solve for *P* (*B*2|*L*).
$$P(B_2|L) = \frac{P(B_2 \text{ and } L \text{ and } D) + P(B_2 \text{ and } L \text{ and } D^C)}{P(L)}$$
= $\frac{P(L|B_2, D)P(B_2|D)P(D) + P(L|B_2, D^C)P(B_2|D^C)P(D^C)}{P(L)}$
= $\f... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.56**",
"token_count": 517,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Expand Equation [2.54](#page-120-0) according to the tree shown in Figure [2.19\(b\),](#page-121-2) and solve for *P* (*B*2|*L*).[38](#page-122-0)
A tree diagram is useful for visualizing the different possible ways that a certain set of outcomes can occur. Although conditional probabilities can certainly be calculat... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**GUIDED PRACTICE 2.57**",
"token_count": 337,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
What is the probability that a white cat has one blue eye and one non-blue eye, given that it is not blind?
Calculate $P(B_1|L^C)$ . Start with the definition of conditional probability, then expand.
$$P(B_1|L^C) = \frac{P(B_1 \text{ and } L^C)}{P(L^C)} = \frac{P(B_1 \text{ and } L^C \text{ and } D) + P(B_1 \text{... | {
"Header 1": "**2.3 Extended example: cat genetics**",
"Header 3": "**EXAMPLE 2.58**",
"token_count": 960,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
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