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H0: The distribution of [...] is the same for each population/treatment. HA: The distribution of [...] is not the same for each population/treatment. We test these hypotheses at the α significance level using a χ2χ2χ2 test for homogeneity. • When there is one random sample and we are looking for association or dependen... |
A news article reports that “Americans have differing views on two potentially inconvenient and invasive practices that airports could implement to uncover potential terrorist attacks.” This news piece was based on a survey conducted among a random sample of 1,137 adults nationwide, where one of the questions on the sur... |
TABLES 355 6.32 Parasitic worm. Lymphatic filariasis is a disease caused by a parasitic worm. Complications of the disease can lead to extreme swelling and other complications. Here we consider results from a randomized experiment that compared three different drug treatment options to clear people of the this parasite,... |
fixed or known population distri- bution; e.g. looking at distribution of color among M&M’s. • χ2χ2χ2 test for homogeneity: – 2 or more independent random samples or randomly allocated treatments – Compare the distribution of a categorical variable across several populations or treatments; e.g. party affiliation over var... |
in the following table: Download No Download Position 1 Position 2 Position 3 13.8% 14.6% 12.1% 18.3% 18.5% 22.7% (a) Calculate the actual number of site visitors in each of the six response categories. (b) Each individual in the experiment had an equal chance of being in any of the three experiment groups. However, w... |
hypothesis test. 46Pew Research Center Publications, Civil War at 150: Still Relevant, Still Divisive, data collected between March 30 - April 3, 2011. 358 CHAPTER 6. INFERENCE FOR CATEGORICAL DATA 6.37 College smokers. who smoke. Out of a random sample of 200 students from this university, 40 students smoke. We are i... |
. (d) What does “95% confidence” mean? (e) Now calculate a 99% confidence interval for the same parameter and interpret it in the context of the data. (f) Compare the widths of the 95% and 99% confidence intervals. Which one is wider? Explain. 6.40 Diabetes and unemployment. A Gallup poll surveyed Americans about their em... |
012 randomly sampled Americans agree with this decision. At a 95% confidence level, this sample has a 3% margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning.48 (a) We are 95% confident that between 43% and 49% of Americans in this sample support ... |
≤ 10 minutes 11-30 minutes 31-60 minutes > 60 minutes 198 130 48 29 6.45 Which chi-square test? Part 2. Consider each of the following planned studies. Determine (i) if a goodness of fit test, test for homogeneity, or test for independence is more appropriate, and (ii) how many degrees of freedom should be used for the... |
this chapter, we focus on inference procedures for numerical data and we encounter a new distribution. In each case, the inference ideas remain the same: 1. Determine which point estimate or test statistic is useful. 2. Identify an appropriate distribution for the point estimate or test statistic. 3. Apply the ideas f... |
deviation to find a Z-score. However, in the case of inference, these values will be unknown. In rare circumstances we may know the standard deviation of a population, even though we do not know its mean. For example, in some industrial processes, the mean may be known to shift over time, while the standard deviation o... |
distribution is known as the t-distribution. A t-distribution, shown as a solid line in Figure 7.1, has a bell shape. However, its tails are thicker than the normal model’s. We can see that a greater proportion of the area under the t-distribution is beyond 2 standard units from 0 than under the normal distribution. T... |
... 1.646 1.645 90% 0.025 12.71 4.303 3.182... 2.110 2.101 2.093 2.086... 1.962 1.960 95% 0.010 31.82 6.965 4.541... 2.567 2.552 2.539 2.528... 2.330 2.326 98% 0.005 63.66 9.925 5.841 2.898 2.878 2.861 2.845 2.581 2.576 99% Confidence level C Figure 7.3: An abbreviated look at the t-table. Each row represents a different... |
degrees of freedom, what percent of the curve is contained between -1.330 and +1.330? Using row df = 18, we find 1.330 in the table. The area in each tail is 0.100 for a total of 0.200, which leaves 0.800 in the middle between -1.33 and +1.33. This corresponds to the 80%, which can be found at the very bottom of that c... |
a calculator or statistical software to get a precise answer. −4−2024−4−2024−4−2024 366 CHAPTER 7. INFERENCE FOR NUMERICAL DATA 7.1.3 Technology: finding area under the ttt-distribution It is possible to find areas under a t-distribution on a calculator. TI-84: FINDING AREA UNDER THE T-CURVE Use 2ND VARS, tcdf to find an... |
to the right of t = 3 with 35 degrees of freedom?2 1Because we want to shade to the right of t = 3, we let lower = 3. There is no upper bound, so use a large value such as 100 for upper. Let df = 35. The area is 0.0025 or 0.25%. 2Because the t-distribution has greater spread and thicker tails than the normal distribut... |
distribution is nearly normal. However, the data can suggest to us whether the population distribution being nearly normal is an unreasonable assumption. THE NORMALITY CONDITION WITH SMALL SAMPLES If the sample is small and there is strong skew or extreme outliers in the data, the population from which the sample was ... |
photos.com). CC BY 2.0 license. n 19 ¯x 4.4 s minimum maximum 2.3 1.7 9.2 Figure 7.7: Summary of mercury content in the muscle of 19 Risso’s dolphins from the Taiji area. Measurements are in µg/wet g (micrograms of mercury per wet gram of muscle). With both conditions met, we will construct a 95% confidence interval. Re... |
�dence interval for the average mercury content in dolphin muscles. point estimate ± critical value × SE of estimate s √ n 2.3 √ 19 4.4 ± 2.10 × ¯x ± t × df = n − 1 df = 18 = (3.29, 5.51) EXAMPLE 7.8 How do we interpret this 95% confidence interval? To what population is it applicable? A random sample of Risso’s dolphin... |
dence level C% (, ) Conclude: Interpret the interval and, if applicable, draw a conclusion in context.. A conclusion Here, we are C% confident that the true mean of [...] is between depends upon whether the interval is entirely above, is entirely below, or contains the value of interest. and 7.1. INFERENCE FOR A MEAN WI... |
and confidence level C%. For a 95% confidence level and df = 15 − 1 = 14, t = 2.145. So the 95% confidence interval is given by: 0.287 ± 2.145 × 0.069 √ 15 df = 14 0.287 ± 2.145 × 0.0178 = (0.249, 0.325) Conclude: We are 95% confident that the true average mercury content of croaker white fish (Pacific) is between 0.249 and... |
data into a list. 3. Choose the INTR option (F3 button), t (F2 button), and 1-S (F1 button). 4. Choose either the Var option (F2) or enter the data in using the List option. 5. Specify the interval details: • Confidence level of interest for C-Level. • If using the Var option, enter the summary statistics. If using Lis... |
population standard deviation and the desired n will be large, so we can use M E = z × σ√ n, making it easier to solve for n. EXAMPLE 7.12 Blood pressure oscillates with the beating of the heart, and the systolic pressure is defined as the peak pressure when a person is at rest. The standard deviation of systolic blood... |
tend to run slower. In fact, all of these components might be influencing run time. We consider this question in the context of the Cherry Blossom Race, which is a 10-mile race in Washington, DC each spring. The average time for all runners who finished the Cherry Blossom Race in 2006 was 93.3 minutes (93 minutes and ab... |
test statistic and p-value for the hypothesis test. Since we will be using a sample standard deviation in our calculation of the test statistic, we will need to use a t-distribution, just as we did with confidence intervals for a mean. We call the test statistic a T -statistic. It has the same general form as a Z-stati... |
it to be outside the 95% confidence interval. However, because the hypothesized value of 93.3 was not rejected by the two-sided α = 0.01 test, we would expect it to fall inside the (wider) 99% confidence interval. 376 CHAPTER 7. INFERENCE FOR NUMERICAL DATA HYPOTHESIS TEST FOR A MEAN To carry out a complete hypothesis t... |
15, a sample mean and standard deviation were computed as 0.287 and 0.069 ppm (parts per million), respectively. Carry out an appropriate test to determine if 0.25 is a reasonable value for the average mercury content of croaker white fish (Pacific). Use the five step method to organize your work. Identify: We will test ... |
interval for the average mercury content in croaker white fish was (0.249, 0.325). Discuss whether the conclusion of the hypothesis test in the previous example is consistent or inconsistent with the conclusion of the confidence interval.4 4It is consistent because 0.25 is located (just barely) inside the confidence inte... |
ify the test details: • Specify the sidedness of the test using the F1, F2, and F3 keys. • Enter the null value, µ0. • If using the Var option, enter the summary statistics. If using List, specify the list and leave Freq values at 1. 8. Hit the EXE button, which returns alternative hypothesis ¯x sx n t T-statistic p-va... |
test require that the sampling distribution for ¯x be nearly normal. For this reason we must check that the following conditions are met. 1. Independence: The data come from a random sample or random process. When sampling without replacement, check that the sample size is less than 10% of the population size. 2. Large... |
1 degree of freedom. Determine which is which, and explain your reasoning. 7.3 Find the p-value, Part I. A random sample is selected from an approximately normal population with an unknown standard deviation. Find the p-value for the given sample size and test statistic. Also determine if the null hypothesis would be ... |
hours or more than 8 hours. n 25 ¯x 7.73 s min max 9.78 6.17 0.77 (a) Write the hypotheses in symbols and in words. (b) Check conditions, then calculate the test statistic, T, and the associated degrees of freedom. (c) Find and interpret the p-value in this context. Drawing a picture may be helpful. (d) What is the co... |
the p-value be equal to 0.05? Assume that all conditions necessary for inference are satisfied. 7.10 ttt vs. zzz. For a given confidence level, t than z∗ affects the width of the confidence interval. df is larger than z. Explain how t∗ df being slightly larger Georgianna claims that in a small city renowned for its music ... |
wants to collect data such that he can get a margin of error of no more than $10 at a 95% confidence level. How large of a sample should he collect? 7.14 SAT scores. The standard deviation of SAT scores for students at a particular Ivy League college is 250 points. Two statistics students, Raina and Luke, want to estim... |
those, 68 required books that could be found on Amazon. A portion of the data set from these courses is shown in Figure 7.9, where prices are in U.S. dollars. subject course number 1 American Indian Studies M10 2 Anthropology 3 Arts and Architecture...... 67 Korean 68 2 10... 1 M10 Jewish Studies bookstore 47.97 14.26... |
can use the same t- distribution techniques we applied in the last section. ndiff 68 ¯xdiff 3.58 sdiff 13.42 Figure 7.11: Summary statistics for the price differences. There were 68 books, so there are 68 differences. 8The difference is taken as UCLA Bookstore price − Amazon price. Because the difference is positive, it tell... |
. Here, the null hypothesis is that the true mean of the differences is 0. T = point estimate − null value SE of estimate = 3.58 − 0 1.63 = 2.20 The degrees of freedom are df = 68 − 1 = 67. To visualize the p-value, the sampling distribution for ¯xdiff is drawn as though H0 is true. This is shown in Figure 7.12. Because ... |
erences µdiff is equal to 0, Identify: Identify the hypotheses and the significance level, α. H0: µdiff = 0 HA: µdiff = 0; HA: µdiff > 0; or HA: µdiff < 0 Choose: Choose the appropriate test procedure and identify it by name. To test hypotheses about a mean of differences we use a 1-sample ttt-test with paired data. Check: Ch... |
students took the SAT before and after taking the company’s course, so we have a difference in scores for each student. We will examine these differences x1 = 57, x2 = 133,..., x30 = 140. The distribution of the differences has a mean of 135.9, a standard deviation of 82.2, and is shown below. Do the data provide convinc... |
�s claim that students’ scores improve by more than 100 points, on average, following the class. GUIDED PRACTICE 7.23 Because we found evidence to support the company’s claim, does this mean that a student will score more than 100 points higher on the SAT if they take the class than if they do not take the class?9 9No.... |
�erences, follow the instructions for a 1-sample t-test. 7.2.4 Confidence intervals for the mean of a difference In the previous examples, we carried out a 1-sample t-test with paired data, where the null hypothesis was that the true mean of differences is zero. Sometimes we want to estimate the true mean of differences w... |
more expensive the UCLA bookstore might be, on average. EXAMPLE 7.24 Based on the interval, can we say that 95% of the books cost between $0.33 and $6.83 more at the UCLA Bookstore than on Amazon? No. This interval is attempting to estimate the average difference with 95% confidence. It is not attempting to capture 95% ... |
�� − 1 and confidence level C% (, ) Conclude: Interpret the interval and, if applicable, draw a conclusion in context. We are C% confident that the true mean of the differences in [...] and. If applicable, draw a conclusion based on whether the interval is entirely above, is is between entirely below, or contains the valu... |
as follows. point estimate ± t × SE of estimate The point estimate is the sample mean of differences: ¯xdiff = 135.9 The SE of the sample mean of differences is: sdiff√ ndiff = 82.2√ 30 = 15.0 We find t for the one-sample case using the t-table at row df = n − 1 and confidence level C%. For a 95% confidence level and df = 30 ... |
if you have all the data or Stats if you have the mean and standard deviation. • If you choose Data, let List be L3 or the list in which you entered the differences (don’t forget to enter the differences!) and let Freq be 1. • If you choose Stats, enter the mean, SD, and sample size of the differences. 5. Let C-Level be ... |
positive, negative, or zero. For example, the difference of paired data from a matched pairs experiment tells us whether one treatment did better, worse, or the same as the other treatment for each subject. • We use the notation ¯xdiff to represent the mean of the sample differences. Likewise, sdiff is the standard deviat... |
. Confidence interval: point estimate ± t × SE of estimate Test statistic: T = point estimate − null value SE of estimate Here the point estimate is the mean of sample differences: ¯xdiff. The SE of estimate is the SE of a mean of sample differences: sdiff√ ndiff. The degrees of freedom is given by df = ndiff − 1. 7.2. INFERE... |
To find out, we take a random sample of 50 days, and record Intel’s and Southwest’s stock on those same days. (b) We randomly sample 50 items from Target stores and note the price for each. Then we visit Walmart and collect the price for each of those same 50 items. (c) A school board would like to determine whether th... |
in Number of Days−60−40−20020406001020304050−60−40−200204060 394 CHAPTER 7. INFERENCE FOR NUMERICAL DATA 7.20 High School and Beyond, Part I. The National Center of Education Statistics conducted a survey of high school seniors, collecting test data on reading, writing, and several other subjects. Here we examine a si... |
scores of a random sample of 200 students who took the High School and Beyond Survey in Exercise 7.20. The mean and standard deviation of the differences are ¯xread−write = −0.545 and 8.887 points. (a) Calculate a 95% confidence interval for the average difference between the reading and writing scores of all students. (... |
�erence is zero or not. Before we perform inference for the difference of means, let’s review the sampling distribution for ¯x1 − ¯x2, which will be used as the point estimate for µ1 − µ2. We know from Section 4.3 that when the independence condition is satisfied, the sampling distribution for ¯x1 − ¯x2 is centered on µ1... |
skew or outliers in either group, the assumption that the populations are nearly normal to be reasonable is typically considered reasonable. 7.3. INFERENCE FOR THE DIFFERENCE OF TWO MEANS 397 7.3.3 Confidence intervals for a difference of means What’s in a name? Are employers more likely to offer interviews or higher pa... |
to John was $3,730 higher than the average salary offered to Jennifer. Because there is randomness in which faculty ended up in the John group and which faculty ended up in the Jennifer group, we want to see if the difference of $3,730 is beyond what could be expected by random variation. In order to answer this, we wil... |
PLE 7.30 Verify that conditions are met for a two-sample t-test. Then, construct the 95% confidence interval for the difference of means. We noted previously that this is an experiment and that the two treatments (name Jennifer and name John) were randomly assigned. Also, both sample sizes are well over 30, so the distri... |
��er John more money for the lab manager position than Jennifer. Finding proof of bias for individual cases is a persistent challenge in enforcing anti-discrimination laws. 13Using technology, we get a more precise interval, based on 118.1 df : (1461, 5999). 14A similar study sent out identical resumes with different na... |
table at row df and confidence level C% s2 1 n1 (, ) Conclude: Interpret the interval and, if applicable, draw a conclusion in context.. If We are C% confident that the true difference in mean [...] applicable, draw a conclusion based on whether the interval is entirely above, is entirely below, or contains the value 0. i... |
, we must first find the degrees of freedom. Using a calculator, we find df = 51.9. We round down to 50, and using a t-table at row df = 50 and confidence level 95%, we get t = 2.009. The 95% confidence interval is given by: (79.4 − 74.1) ± 2.009 × 142 30 + 202 30 df = 51.9 5.3 ± 2.009 × 4.46 = (−3.66, 14.26) Conclude: We a... |
PLE T-INTERVAL 1. Navigate to STAT (MENU button, then hit the 2 button or select STAT). 2. If necessary, enter the data into a list. 3. Choose the INTR option (F4 button). 4. Choose the t option (F2 button). 5. Choose the 2-S option (F2 button). 6. Choose either the Var option (F2) or enter the data in using the List o... |
: Four cases from the ncbirths data set. The value “NA”, shown for the first two entries of the first variable, indicates pieces of data that are missing. Figure 7.16: The top panel represents birth weights for infants whose mothers smoked. The bottom panel represents the birth weights for infants whose mothers who did n... |
a t-distribution. mean st. dev. samp. size smoker 6.78 1.43 50 nonsmoker 7.18 1.60 100 Figure 7.17: Summary statistics for the ncbirths data set. EXAMPLE 7.34 We will use the summary statistics in Figure 7.17 for this exercise. (a) What is the point estimate of the population difference, µ1 − µ2? (b) Compute the standa... |
What can we conclude from this p-value? Use a significance level of α = 0.05. This p-value of 0.130 is larger the significance level of 0.05, so we do not reject the null hypothesis. There is not sufficient evidence to say there is a difference in average birth weight of newborns from North Carolina mothers who did smoke d... |
fference of means we use a 2-sample ttt-test. Check: Check conditions for the sampling distribution for ¯x1 − ¯x2 to be nearly normal. 1. Independence: Data come from 2 independent random samples or from a randomized experiment with 2 treatments. When sampling without replacement, check that the sample size is less than... |
ficance level. H0: µ1 − µ2 = 0. The stem cells do not improve heart pumping function. HA: µ1 − µ2 > 0. The stem cells do improve heart pumping function. Choose: Because we are hypothesizing about a difference of means we choose the 2-sample t-test. Check: The data come from a randomized experiment with two treatment grou... |
Choose STAT. 2. Right arrow to TESTS. 3. Choose 4:2-SampTTest. 4. Choose Data if you have all the data or Stats if you have the means and standard deviations. • If you choose Data, let List1 be L1 or the list that contains sample 1 and let List2 be L2 or the list that contains sample 2 (don’t forget to enter the data!... |
heart attack.18 ESCs control n 9 9 ¯x 3.50 -4.33 s 5.17 2.76 18Choose 2-SampTTest or equivalent. Because we have the summary statistics rather than all of the data, choose Stats. Let ¯x1=3.50, Sx1=5.17, n1=9, ¯x2=-4.33, Sx2 = 2.76, and n2 = 9. We get t = 4.01, and the p-value p = 8.4 × 10−4 = 0.00084. The degrees of f... |
1 ≥ 30 and n2 ≥ 30 or both population distribu- tions are nearly normal. - If the sample sizes are less than 30 and it is not known that both population distributions are nearly normal, check for excessive skew or outliers in the data. If neither exists, the condition that both population distributions could be nearly ... |
carats $44.51 $13.32 23 1 carat $56.81 $16.13 23 7.24 Diamonds, Part II. In Exercise 7.23, we discussed diamond prices (standardized by weight) for diamonds with weights 0. 99 carats and 1 carat. See the table for summary statistics, and then construct a 95% confidence interval for the difference in means between the st... |
lll 7.3. INFERENCE FOR THE DIFFERENCE OF TWO MEANS 411 7.26 Fuel efficiency of manual and automatic cars, Part I. Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of car... |
’s need for control or their rebellion against control, and it is part of a commonly used mental health test called the Minnesota Multiphasic Personality Inventory (MMPI) test. The experiment had three treatment groups: (1) Four hours of sensory restriction plus a 15 minute “therapeutic” tape advising that professional... |
val • 1-sample t-test/interval • 1-sample t-test/interval with paired data • 2-sample t-test/interval The above inferential procedures all involve a point estimate, a standard error of the estimate, and an assumption about the shape of the sampling distribution for the point estimate. From Chapter 6, the χ2 tests and t... |
ate 52.1 grams of biscuits, with a standard deviation of 45.1 grams, and patients in the control group ate 27.1 grams of biscuits, with a standard deviation of 26.4 grams. Do these data provide convincing evidence that the average food intake (measured in amount of biscuits consumed) is different for the patients in th... |
��erent than the one from the original sample? Explain your reasoning. (e) The sample means given above are point estimates for the mean number of credits taken by all students at that college. What measures do we use to quantify the variability of this estimate? Compute this quantity using the data from the original s... |
, among many other questions, about the number of exclusive relationships these students have been in. The histogram below shows the distribution of the data from this sample. The sample average is 3.2 with a standard deviation of 1.97. Estimate the average number of exclusive relationships Duke students have been in u... |
: ¯x = 23.44 years old 26Centers for Disease Control and Prevention, National Survey of Family Growth, 2010. Age at first marriage101520253035404502004006008001000 7.3. INFERENCE FOR THE DIFFERENCE OF TWO MEANS 417 7.41 Friday the 13th, Part I. In the early 1990’s, researchers in the UK collected data on traffic flow, nu... |
and Friday the 13th. (b) Calculate a 95% confidence interval for the difference between the average numbers of traffic accident related emergency room admissions between Friday the 6th and Friday the 13th. (c) The conclusion of the original study states, “Friday 13th is unlucky for some. The risk of hospital admission as ... |
fficient and estimate it from a scatterplot. 5. Know and apply the properties of the correlation coefficient. 8.1.1 Fitting a line to data Requests from twelve separate buyers were simultaneously placed with a trading company to purchase Target Corporation stock (ticker TGT, April 26th, 2012). We let x be the number of sto... |
instance, we might wonder, should we move the line up or down a little, or should we tilt it more or less? As we move forward in this chapter, we will learn different criteria for line-fitting, and we will also learn about the uncertainty associated with estimates of model parameters. Figure 8.2: Three data sets where a... |
RELATION 423 The head and total length variables are associated: possums with an above average total length also tend to have above average head lengths. While the relationship is not perfectly linear, it could be helpful to partially explain the connection between these variables with a straight line. We want to descr... |
a small, negative residual of about -1; the observation marked by “+” has a large residual of about +7; and the observation marked by “” has a moderate residual of about -4. The size of a residual is usually discussed in terms of its absolute value. For example, the residual for “” is larger than that of “×” because |... |
�gure, using the linear model: ˆy = 41 + 0.59x.2 Residuals are helpful in evaluating how well a linear model fits a data set. We often display the residuals in a residual plot such as the one shown in Figure 8.7. Here, the residuals are calculated for each x value, and plotted versus x. For instance, the point (85.0, 98... |
the standard deviation of the residuals. The typical error when predicting head length using this model is about 2.5 mm. Figure 8.7: Left: Scatterplot of head length versus total length for 104 brushtail possums. Three particular points have been highlighted. Right: Residual plot for the model shown in left panel. STA... |
ribing linear relationships with correlation When a linear relationship exists between two variables, we can quantify the strength and direction of the linear relation with the correlation coefficient, or just correlation for short. Figure 8.9 shows eight plots and their corresponding correlations. Figure 8.9: Sample sca... |
y corresponds to multiplying all the y values by a certain number. This would change the mean and the standard deviation of y, but it would not change the correlation. To see this, imagine dividing every number on the vertical axis by 10. The units of y are now in cm rather than in mm, but the graph has remain exactly... |
/calculator/paknt6oneh. Figure 8.11: Four scatterplots from Desmos with best fit line drawn in. 8.1. LINE FITTING, RESIDUALS, AND CORRELATION 431 Section summary • In Chapter 2 we introduced a bivariate display called a scatterplot, which shows the relationship between two numerical variables. When we use x to predict y... |
well exist a quadratic or other type of association. – Just like Z-scores, the correlation has no units. Changing the units in which x or y are measured does not affect the correlation. – Correlation is sensitive to outliers. Adding or removing a single point can have a big effect on the correlation. – As we learned pre... |
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lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll(f) 8.1. LINE FITTING, RESIDUALS, AND CORRELATION 433 8.4 Identify relationships, Part II. For each of the six plots, identify the strength of the relationship (e.g. weak, moderate, or strong) in the dat... |
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d) r = −0.85 8.9 Speed and height. 1,302 UCLA students were asked to fill out a survey where they were asked about their height, fastest speed they have ever driven, and gender. The scatterplot on the left displays the relationship between height and fastest speed, and the scatterplot on the right displays the breakdown... |
year and the total rainfall for the year. Eduardo records rainfall in inches and Rosie in centimeters. How will their correlation coefficients compare? 8.11 The Coast Starlight, Part I. The Coast Starlight Amtrak train runs from Seattle to Los Angeles. The scatterplot below displays the distance between each stop (in mi... |
studying anthropometry collected body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender for 507 physically active individuals.7 The scatterplot below shows the relationship between height and shoulder girth (over deltoid muscles), both measured in centimeters. (a) Describ... |
the relevant summary statistics. Interpret these quantities in context. 2. Understand why the least squares regression line is called the least squares regression line. 3. Interpret the explained variance R2. 4. Understand the concept of extrapolation and why it is dangerous. 5. Identify outliers and influential points... |
do data scientists prefer the least squares regression line? One reason is that it is easier to compute by hand and in most statistical software. Another, and more compelling, reason is that in many applications, a residual twice as large as another residual is more than twice as bad. For example, being off by 4 is usu... |
predicting y based on x can be written as: ˆy = a + bx. We first find b, the slope, and then we solve for a, the y-intercept. b = r sy sx ¯y = a + b¯x GUIDED PRACTICE 8.9 Figure 8.14 shows the sample means for the family income and gift aid as $101,800 and $19,940, respectively. Plot the point (101.8, 19.94) on Figure 8... |
− (−5.776)(101.8) = 607.9 = −5.776 ˆy = 607.3 − 5.776x or family income = 607.3 − 5.776 × aid We mentioned earlier that a computer is usually used to compute the least squares line. A summary table based on computer output is shown in Figure 8.15 for the Elmhurst College data. The first column of numbers provides estim... |
cant error in predicting an individual student’s aid. Additionally, the data all come from one freshman class, and the way aid is determined by the university may change from year to year. 8.2.3 Interpreting the coefficients of a regression line Interpreting the coefficients in a regression model is often one of the most ... |
we encountered a data set that compared the price of new textbooks for UCLA courses at the UCLA Bookstore and on Amazon. We fit a linear model for predicting price at UCLA Bookstore from price on Amazon and we get: where x is the price on Amazon and y is the price at the UCLA bookstore. Interpret the coefficients in this... |
Amazon, it costs an extra 1.03 dollars at the UCLA Bookstore. This interpretation does make sense in this context. 10No. The slope describes the overall trend. This is observational data; a causal conclusion cannot be drawn. Remember, a causal relationship can only be concluded by a well-designed randomized, controlle... |
(in the y direction) about the regression line is than the variance about the horizontal line ¯y. For example, consider the Elmhurst College data, shown in Figure 8.16. The variance of the response variable, aid received, is s2 aid = 29.8. However, if we apply our least squares line, then this model reduces our uncert... |
that r = ± Use this fact to answer the next two practice problems. √ R2. GUIDED PRACTICE 8.20 If a linear model has a very strong negative relationship with a correlation of -0.97, how much of the variation in the response variable is explained by the linear model?14 GUIDED PRACTICE 8.21 If a linear model has an R2 or... |
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