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3. 2 • measures of variation 113 variance the variance provides a measure of the spread of data values by using the squared deviations from the mean. the more the individual data values differ from the mean, the larger the variance. a financial advisor might use variance to determine the volatility of an investment and... | openstax_principles-of-data-science-web | [
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can be confusing. by contrast, standard deviation is measured in the same units as the original dataset, and thus the standard deviation is more commonly used to measure the spread of a dataset. standard deviation the standard deviation of a dataset provides a numerical measure of the overall amount of variation in a d... | openstax_principles-of-data-science-web | [
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. this means that once the sample variance has been calculated, the sample standard deviation can then be easily calculated as the square root of the sample variance, as in example 3. 7. example 3. 7 problem a biologist calculates that the sample variance for the amount of plant growth for a sample of plants is 8. 7 cm... | openstax_principles-of-data-science-web | [
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3. 2 • measures of variation 115 solution the sample standard deviation ( ) is calculated as the square root of the variance. this result indicates that the standard deviation is about 6. 5 years. notice that the sample variance is the square of the sample standard deviation, so if the sample standard deviation is know... | openstax_principles-of-data-science-web | [
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3. 3 measures of position learning outcomes by the end of this section, you should be able to : • 3. 3. 1 define and calculate percentiles, quartiles, and - scores for a dataset. • 3. 3. 2 use python to calculate measures of position for a dataset. common measures of position include percentiles and quartiles as well a... | openstax_principles-of-data-science-web | [
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than your score. percentiles are useful for comparing many types of values. for example, a stock market mutual fund might report that the performance for the fund over the past year was in the 80th percentile of all mutual funds in the peer group. this indicates that the fund performed better than 80 % of all other fun... | openstax_principles-of-data-science-web | [
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3. 3 • measures of position 117 example 3. 10 problem the following ordered dataset represents the scores of 15 employees on an aptitude test : 51, 63, 65, 68, 71, 75, 75, 77, 79, 82, 88, 89, 89, 92, 95 determine the percentile for the employee who scored 88 on the aptitude test. solution there are 15 data values in to... | openstax_principles-of-data-science-web | [
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. 1. note that these are the data values below the median. the upper half of the data values are 7. 4, 7. 5, 7. 9, 8. 2, 8. 7, which are the data values above the median. to find the first quartile,, locate the middle value of the lower half of the data ( 5. 4, 6. 0, 6. 3, 6. 8, 7. 1 ). the middle value of the lower ha... | openstax_principles-of-data-science-web | [
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a dataset. for example, if most employees at a company earn about $ 50, 000 and the ceo of the company earns $ 2. 5 million, then we consider the ceo ’ s salary to be an outlier data value because this salary is significantly different from all the other salaries in the dataset. an outlier data value can also be a valu... | openstax_principles-of-data-science-web | [
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13 data values, so the median is the middle data value, which is 488, 800. next, calculate the and. for the first quartile, look at the data values below the median. the two middle data values in this lower half of the data are 230, 500 and 387, 000. to determine the first quartile, find the mean of these two data valu... | openstax_principles-of-data-science-web | [
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3. 3 • measures of position 119 now, calculate the interquartile range ( iqr ) : calculate the value of 1. 5 interquartile range ( iqr ) : calculate the lower and upper bound for outliers : the lower bound for outliers is −201, 625. of course, no home price is less than −201, 625, so no outliers are present for the low... | openstax_principles-of-data-science-web | [
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will be a negative value. if the measurement is exactly equal to the mean, the corresponding - score will be zero. if the measurement is above the mean, the corresponding - score will be a positive value. - scores can also be used to identify outliers. since - scores measure the number of standard deviations from the m... | openstax_principles-of-data-science-web | [
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3. 4 probability theory learning outcomes by the end of this section, you should be able to : • 3. 4. 1 describe the basic concepts of probability and apply these concepts to real - world applications in data science. • 3. 4. 2 apply conditional probability and bayes ’ theorem. probability is a numerical measure that a... | openstax_principles-of-data-science-web | [
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3. 4 • probability theory 121 want to know the probability of rain. the probability of obtaining heads on one flip of a coin is one - half, or 0. 5. a data scientist is in interested in expressing probability as a number between 0 and 1 ( inclusive ), where 0 indicates impossibility ( the event will not occur ) and 1 i... | openstax_principles-of-data-science-web | [
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school funding. the probability for a randomly selected person being in favor of increased funding can then be calculated as follows ( notice that event in this example corresponds to the event that a person is in favor of the increased funding ) : example 3. 15 problem a medical patient is told they need knee surgery,... | openstax_principles-of-data-science-web | [
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randomly guesses. what is the probability that the student selects the correct answer? solution since the student is guessing, each answer choice is equally likely to be selected. there is 1 correct answer out of 5 possible choices. the probability of selecting the correct answer can be calculated as : notice in exampl... | openstax_principles-of-data-science-web | [
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3. 4 • probability theory 123 example 3. 17 problem a company estimates that the probability that an employee will provide confidential information to a hacker is 0. 1 %. determine the probability that an employee will not provide any confidential information during a hacking attempt. solution let event be the event th... | openstax_principles-of-data-science-web | [
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of one event affects the probability of occurrence of another event. independent events are events where the probability of occurrence of one event is not affected by the occurrent of another event. the dependence of events has important implications in many fields such as marketing, engineering, psychology, and medici... | openstax_principles-of-data-science-web | [
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affected by the selection of the ace, so these events are dependent. there are several ways to use conditional probabilities in data science applications. conditional probability can be defined as follows : when assessing the conditional probability of, if the two events are independent, this indicates that event is no... | openstax_principles-of-data-science-web | [
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3. 4 • probability theory 125 age group nursing degrees non - nursing degrees total 22 and under 1036 1287 2323 23 and older 986 932 1918 total 2022 2219 4241 table 3. 2 number of nursing and non - nursing degrees at a university by age group solution since we are given that the group of interest are those graduates in... | openstax_principles-of-data-science-web | [
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in many data science applications. for example, a doctor might be interested to know the probability that at least one surgery to be performed this week will involve an infection of some type. the phrase “ at least one ” implies the condition of one or more successes. from a sample space perspective, one or more succes... | openstax_principles-of-data-science-web | [
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chance that all four knee surgeries will be successful. 2. the probability that a knee surgery will be unsuccessful can be calculated using the complement rule. if the probability of a successful surgery is 0. 89, then the probability that the surgery will be unsuccessful is 0. 11 : based on this, the probability that ... | openstax_principles-of-data-science-web | [
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3. 4 • probability theory 127 since this is a very small probability, it is very unlikely that none of the surgeries will be successful. 3. to calculate the probability that at least one of the knee surgeries will be successful, use the probability formula for “ at least one, ” which is calculated as the complement of ... | openstax_principles-of-data-science-web | [
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result. the researcher is interested in calculating, where event is the person actually has cancer and event is the event that the person shows a positive result in the screening test. use bayes ’ theorem to calculate this conditional probability. 128 3 • descriptive statistics : statistical measurements and probabilit... | openstax_principles-of-data-science-web | [
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3. 5 discrete and continuous probability distributions learning outcomes by the end of this section, you should be able to : • 3. 5. 1 describe fundamental aspects of probability distributions. • 3. 5. 2 apply discrete probability distributions including binomial and poisson distributions. • 3. 5. 3 apply continuous pr... | openstax_principles-of-data-science-web | [
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, the number of values is countable if it possible to count them individually. ) typically, a discrete random variable is the result of a count of some kind. for example, if the random variable represents the number of cars in a | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 129 parking lot, then the values that x can take on can only be whole numbers since it would not make sense to have cars in the parking lot. • continuous random variable — a random variable is considered continuous if the value of the random variable can take on ... | openstax_principles-of-data-science-web | [
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or continuous random variables : 1. the amount of gas, in gallons, used to fill a gas tank 2. number of children per household in a certain neighborhood 130 3 • descriptive statistics : statistical measurements and probability distributions access for free at openstax. org 3. number of text messages sent by a certain s... | openstax_principles-of-data-science-web | [
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and the two possible outcomes can be considered as success or failure. for example, when a baseball player is at - bat, the player either gets a hit or does not get a hit. there are many applications of binomial experiments that occur in medicine, psychology, engineering, science, marketing, and other fields. there are... | openstax_principles-of-data-science-web | [
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the experiment. for example, from past data we know that 35 % of people prefer vanilla as their favorite ice cream flavor. if a group of 15 individuals are surveyed to ask if vanilla is their favorite ice cream flavor, the probability of success for each trial will be 0. 35. • the random variable will count the number ... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 131 the number of successes, this implies that will be a discrete random variable. for example, if the researcher is counting the number of people in the group of 15 that respond to say vanilla is their favorite ice cream flavor, then can take on values such as 3... | openstax_principles-of-data-science-web | [
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surgery is 92 %, so. the number of successes of interest is 18 since the researcher wants to determine the probability that 18 of the 20 patients had a successful result from the surgery, so. when calculating the probability for successes in a binomial experiment, a binomial probability formula can be used, but in many... | openstax_principles-of-data-science-web | [
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than 14 are not shown on the graph since these corresponding probabilities are very close to zero. figure 3. 5 graph of the binomial distribution for and since these computations tend to be complicated and time - consuming, most data scientists will use technology ( such as python, r, excel, or others ) to calculate bi... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 133 poisson distribution the goal of a binomial experiment is to calculate the probability of a certain number of successes in a specific number of trials. however, there are certain scenarios where a data scientist might be interested to know the probability of ... | openstax_principles-of-data-science-web | [
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variable will be discrete. to calculate the probability of successes, the poisson probability formula can be used, as follows : where : is the average or mean number of occurrences per interval is the constant 2. 71828 … example 3. 26 problem from past data, a traffic engineer determines the mean number of vehicles ent... | openstax_principles-of-data-science-web | [
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many values. we used the example about that if the random variable represents the weight of a bag of apples, then can take on any value such as pounds of apples. many probability distributions apply to continuous random variables. these distributions rely on determining the probability that the random variable falls wi... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 135 in this section, we will examine an important continuous probability distribution that relies on the probability density function, namely the normal distribution. many variables, such as heights, weights, salaries, and blood pressure measurements, follow a no... | openstax_principles-of-data-science-web | [
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symmetric about. as the notation indicates, the normal distribution depends only on the mean and the standard deviation. because the area under the curve must equal 1, a change in the standard deviation,, causes a change in the shape of the normal curve ; the curve becomes fatter and wider or skinnier and taller depend... | openstax_principles-of-data-science-web | [
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the area under the normal probability density curve to the right of the data value of $ 68, 000. see using python with probability distributions for the specific python program and results. the resulting probability is calculated as 0. 143. thus, there is a probability of about 14 % that a random employee has a salary ... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 137 of the mean ). • about 95 % of the - values lie between and units from the mean ( within two standard deviations of the mean ). • about 99. 7 % of the - values lie between and units from the mean ( within three standard deviations of the mean ). notice that a... | openstax_principles-of-data-science-web | [
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. these functions are part of a library called scipy. stats ( https : / / openstax. org / r / scipy ). here are a few of these probability density functions available within python : • binom ( ) — calculate probabilities associated with the binomial distribution • poisson ( ) — calculate probabilities associated with t... | openstax_principles-of-data-science-web | [
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: binom. pmf ( 18, 20, 0. 92 ) the round ( ) function is then used to round the probability result to 3 decimal places. here is the input and output of this python program : # import the binom function from the scipy. stats library from scipy. stats import binom # define parameters x, n, and p : x = 18 n = 20 p = 0. 92... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 139 using python with the normal distribution the norm ( ) function in python allows calculations of normal probabilities. the probability density function is sometimes called the cumulative density function, and so this is referred to as norm. cdf ( ) within pyt... | openstax_principles-of-data-science-web | [
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normal probability - note this is # the area to the left # subtract this result from 1 to obtain area to the right of the x - value # use round ( ) function to round answer to 3 decimal places round ( 1 - norm. cdf ( x, mean, standard _ deviation ), 3 ) the resulting output will look like this : python code 140 3 • des... | openstax_principles-of-data-science-web | [
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3. 5 • discrete and continuous probability distributions 141 key terms bayes ’ theorem a method used to calculate a conditional probability when additional information is obtained to refine a probability estimate binomial distribution a probability distribution for a discrete random variable that is the number of succe... | openstax_principles-of-data-science-web | [
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not affected by the occurrence of another event interquartile range ( iqr ) a number that indicates the spread of the middle half, or middle 50 %, of the data ; the difference between the third quartile ( ) and the first quartile ( ) mean ( also called arithmetic mean ) a measure of center of a dataset, calculated by a... | openstax_principles-of-data-science-web | [
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distribution of a discrete random variable quartiles numbers that divide an ordered dataset into quarters ; the second quartile is the same as the median random variable a variable where a single numerical value is assigned to a specific outcome from an experiment range a measure of dispersion for a dataset calculated ... | openstax_principles-of-data-science-web | [
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bureau of labor statistics ( https : / / openstax. org / r / bls ). as a group : • research salary data by gender for a certain year. • compile descriptive statistics where available for statistical measurements including the mean, median, quartiles, and standard deviation by gender. • create graphical presentations fo... | openstax_principles-of-data-science-web | [
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reliever to provide relief from migraine headache ( time in minutes ) sample of 10 patients : 12. 1, 13. 8, 9. 4, 15. 9, 11. 5, 14. 6, 18. 1, 12. 7, 11. 0, 14. 2 a. for dataset c, calculate the range, standard deviation, and variance ( round answers to 1 decimal place ). use technology as appropriate and include approp... | openstax_principles-of-data-science-web | [
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. 6. at a website development company, the average number of sick days taken by employees per year is 9 days with a standard deviation of 2. 3 days. a. a certain employee has taken 2 sick days for the year. calculate the - score for this employee ( round your answer to 2 decimal places ). b. what does the - score commu... | openstax_principles-of-data-science-web | [
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older ) get the flu each year. • 32 % of people under 65 get the flu each year. • in the population, there are 15 % senior citizens. a. are the events “ being a senior ” and “ getting the flu ” dependent or independent? b. find the probability that a person selected at random is a senior and will get the flu. c. find t... | openstax_principles-of-data-science-web | [
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. b. find the probability that a randomly selected cell phone customer has a monthly bill more than $ 85. 146 3 • quantitative problems access for free at openstax. org figure 4. 1 inferential statistics is used extensively in data science to draw conclusions about a larger population — and drive decision - making. ( c... | openstax_principles-of-data-science-web | [
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4. 4 analysis of variance ( anova ) introduction inferential statistics plays a key role in data science applications, as its techniques allow researchers to infer or generalize observations from samples to the larger population from which they were selected. if the researcher had access to a full set of population dat... | openstax_principles-of-data-science-web | [
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applications in data science, including in machine learning models where a mathematical model is created to determine a relationship between input and output variables of a dataset. several such applications of regression analysis in machine learning are further explored in decision - making using machine learning basi... | openstax_principles-of-data-science-web | [
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4. 1 statistical inference and confidence intervals learning outcomes by the end of this section, you should be able to : • 4. 1. 1 estimate parameters, create confidence intervals, and calculate sample size requirements. • 4. 1. 2 apply bootstrapping methods for parameter estimation. • 4. 1. 3 use python to calculate ... | openstax_principles-of-data-science-web | [
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collect data from a sample to make inferences about a population, they calculate a point estimate based on the observed sample data. ( see survey design and implementation for coverage of sampling techniques. ) the point estimate serves as the best guess or approximation for the parameter's actual value. a confidence i... | openstax_principles-of-data-science-web | [
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• the margin of error as noted, the point estimate is a single number that is used to estimate the population parameter. the margin of error ( usually denoted by e ) provides an indication of the maximum error of the estimate. the margin of error can be viewed as the maximum distance around the point estimate where the... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 149 the data scientist can then state the following conclusion : there is 95 % confidence that the forecast for median income for all residents of california is between $ 64, 000 and $ 73, 000. example 4. 1 problem a medical researcher is interested in estimating th... | openstax_principles-of-data-science-web | [
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of the sample size. a. written mathematically : the central limit theorem describes the relationship between the sample distribution of sample means and 150 4 • inferential statistics and regression analysis access for free at openstax. org the underlying population. this theorem is an important tool that allows data s... | openstax_principles-of-data-science-web | [
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the sample size is at least 30, or the underlying population is known to follow a normal distribution. c. the population standard deviation ( ) is known. once these conditions are met, the margin of error is calculated according to the following formula : where : is called the critical value of the normal distribution ... | openstax_principles-of-data-science-web | [
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2. 575 table 4. 2 typical values of for various confidence levels graphically, the critical values can be marked on the normal distribution curve, as shown in figure 4. 2. ( see discrete and continuous probability distributions for a review of the normal distribution curve. ) figure 4. 2 is an example for a 95 % confid... | openstax_principles-of-data-science-web | [
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6. 3 hours. the professor would like to forecast the amount of time spent on homework in future semesters. create a forecasted confidence interval using both a 90 % and 95 % confidence interval and provide a conclusion regarding the confidence interval. also compare the widths of the two confidence intervals. which con... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 153 parameters held constant, we should expect that the confidence interval will become wider. another way to consider this : the wider the confidence interval, the more likely the interval is to contain the true population mean. this makes intuitive sense in that i... | openstax_principles-of-data-science-web | [
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3 comparison of t - distribution and normal distribution curves the t - distribution is actually a family of curves, determined by a parameter called degrees of freedom ( df ), where df is equal to. the critical value is similar to a z - score and specifies the area under the t - distribution curve corresponding to the... | openstax_principles-of-data-science-web | [
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2. 750 table 4. 3 typical values of for various confidence levels and degrees of freedom note that python can be used to calculate these critical values. python provides a function called t. ppf ( ) that generates the value of the t - distribution corresponding to a specified area under the t - distribution curve and s... | openstax_principles-of-data-science-web | [
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2. 131 once the margin of error is calculated, the confidence interval is formed in the same way as the previous section, namely : example 4. 4 problem a company ’ s human resource administrator wants to estimate the average commuting distance for all 5, 000 employees at the company. since it is impractical to collect ... | openstax_principles-of-data-science-web | [
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a researcher might be interested in the proportion of smokers among u. s. adults, and the number of smokers would be considered the number of successes. some terminology will be helpful : represents the population proportion, which is typically unknown. represents the sample proportion. represents the number of success... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 157 start off by calculating the sample proportion : verify that the normal approximation to the binomial distribution is appropriate by ensuring that both and are both at least 5, where represents the sample proportion. for this example,, and. both of these results... | openstax_principles-of-data-science-web | [
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is the critical value of the normal distribution. is the population standard deviation. is the desired margin of error. note that for sample size calculations, sample size results are rounded up to the next higher whole number. 158 4 • inferential statistics and regression analysis access for free at openstax. org for ... | openstax_principles-of-data-science-web | [
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estimate for the sample proportion is available, then that prior estimate should be utilized. • if a prior estimate for the sample proportion is not available, then use. | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 159 example 4. 7 problem political candidate smith is planning a survey to determine a 95 % confidence interval for the proportion of voters who plan to vote for smith. how many people should be surveyed? assume a margin of error is 3 %. a. assume there is no prior ... | openstax_principles-of-data-science-web | [
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do when these requirements are not met? fortunately, there is another option called “ bootstrapping ” that can be used to find confidence intervals when the underlying distribution is unknown or if one of the conditions is not met. this bootstrapping method involves repeatedly taking samples with replacement. sampling ... | openstax_principles-of-data-science-web | [
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from which the data are sampled. ( parametric methods, by contrast, assume a specific form for the underlying distribution and require estimating parameters. ) since bootstrapping requires a large number of repeated samples, software ( such as excel, python, or r ) is often used to automate the repetitive sampling proc... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 161 student id student age 005 29 006 23 007 21 008 20 009 19 010 21 011 25 012 28 013 22 014 37 015 24 016 31 017 23 018 19 019 26 020 20 table 4. 4 ages of 20 randomly selected students from one college solution for the bootstrapping process, we will form samples ... | openstax_principles-of-data-science-web | [
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by 2. for this example, add 21. 9 to 22. 2 and divide by 2. the result is 22. 05. to find the 95th percentile ( p95 ), multiply the percentile ( 95 % ) times the number of data values, which is 20. the result is 19, so to find the 95th percentile, add the 19th and 20th data values together and divide by 2. for this exa... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 163 # define random sample of ages ages = [ 22, 23, 25, 31, 24, 21, 28, 23, 21, 20, 22, 19, 34, 19, 37, 26, 29, 21, 24, 20 ] # convert ages to sequence ages = ( ages, ) # use bootstrap function for confidence interval for the mean conf _ interval = bootstrap ( ages,... | openstax_principles-of-data-science-web | [
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with these calculations. table 4. 5 provides a summary of various functions available within the scipy library ( https : / / openstax. org / r / spy ) for confidence interval calculations : 164 4 • inferential statistics and regression analysis access for free at openstax. org usage python function name syntax calculat... | openstax_principles-of-data-science-web | [
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4. 1 • statistical inference and confidence intervals 165 # confidence level confidence _ level = 0. 90 # standard error standard _ error = population _ standard _ deviation / math. sqrt ( sample _ size ) # calculate confidence interval using norm. interval function stats. norm. interval ( confidence _ level, sample _ ... | openstax_principles-of-data-science-web | [
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4. 2 hypothesis testing learning outcomes by the end of this section, you should be able to : • 4. 2. 1 apply hypothesis testing methods to test statistical claims involving one sample. • 4. 2. 2 use python to assist with hypothesis testing calculations. • 4. 2. 3 conduct hypothesis tests to compare two means, two prop... | openstax_principles-of-data-science-web | [
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the null hypothesis. it provides the foundation for various data science investigations and plays a role at different stages of the analyses such as data collection and validation, modeling related tasks, and determination of statistical significance. for example, a local restaurant might claim that the average deliver... | openstax_principles-of-data-science-web | [
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4. 2 • hypothesis testing 167 the steps for hypothesis testing are as follows : 1. set up a null and alternative hypothesis based on the claim. identify whether the null or the alternative hypothesis represents the claim. 2. collect relevant sample data to investigate the claim. 3. determine the correct distribution to... | openstax_principles-of-data-science-web | [
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equals symbol, then the alternative hypothesis will contain a not equals symbol. if the null hypothesis contains a symbol, then the alternative hypothesis will contain a symbol. if the null hypothesis contains a symbol, then the alternative hypothesis will contain a symbol. note that the alternative hypothesis will alw... | openstax_principles-of-data-science-web | [
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##s to use for a given hypothesis test : 1. determine if the hypothesis test involves a claim for a population mean or population proportion. if the hypothesis test involves a population mean, use either setup a, b, or c. 2. if the hypothesis test involves a population proportion, use either setup d, e, or f. 3. transl... | openstax_principles-of-data-science-web | [
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4. 2 • hypothesis testing 169 example 4. 12 problem write the null and alternative hypotheses for the following claim : a medical researcher claims that the proportion of adults in the united states who are smokers is at most 25 %. solution note that the claim involves the phrase “ at most, ” which translates to a less... | openstax_principles-of-data-science-web | [
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is called the level of significance, denoted by the greek letter alpha,. from a practical standpoint, the level of significance is the probability value used to determine when the sample data indicates significant evidence against the null hypothesis. this level of significance is typically set to a small value, which ... | openstax_principles-of-data-science-web | [
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distribution curve. since the p - value is a probability, any p - value must always be a numerical value between 0 and 1 inclusive. when calculating a p - value as the area under the probability distribution curve, the corresponding area will be determined using the location under the curve, which favors the rejection ... | openstax_principles-of-data-science-web | [
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“ fail to reject the null hypothesis ” might be difficult to interpret for those not familiar with the terminology of hypothesis testing. these decisions can be translated into concluding statements such as those shown in table 4. 8. select the decision from the hypothesis test : is the claim in the null or the alterna... | openstax_principles-of-data-science-web | [
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4. 2 • hypothesis testing 171 testing claims for the mean when the population standard deviation is known recall that in the discussion for confidence intervals we examined the confidence interval for the population mean when the population standard deviation is known ( normal distribution is used ) or when the populat... | openstax_principles-of-data-science-web | [
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to a greater than or equals symbol. a greater than or equals symbol must be used in the null hypothesis. the complement of greater than or equals is a less than symbol, which will be used in the alternative hypothesis. the claim refers to the population mean time that college students spend on social media, so the symb... | openstax_principles-of-data-science-web | [
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stats import norm from scipy. stats import norm # define parameters x, mean and standard _ deviation : x = - 1. 846 mean = 0 standard _ deviation = 1 # use norm. cdf function to calculate normal probability - note this is the area to the left of the x - value # subtract this result from 1 to obtain area to the right of... | openstax_principles-of-data-science-web | [
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4. 2 • hypothesis testing 173 hypothesis. this is accomplished by comparing the p - value with the level of significance. in this example, the p - value is 0. 032 and the level of significance is 0. 05 : since the p - value level of significance, then the decision is to “ reject the null hypothesis. ” conclusion : the ... | openstax_principles-of-data-science-web | [
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collects a sample of 50 smartphones and determines that the mean battery life of the sample is 24. 1 hours with a sample standard deviation of 4. 1 hours. use this sample data to test the claim made by the smartphone manufacturer. use a level of significance of 0. 10 for this analysis. solution the first step in the hy... | openstax_principles-of-data-science-web | [
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areas, which is 0. 127. the area under the t - distribution curve can be found using software. for example, in python, the area under the t - distribution curve to the left of a t - score of can be obtained using the function t. cdf ( ) the syntax for using this function is t. cdf ( x, df ) where : x is the measurement... | openstax_principles-of-data-science-web | [
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0. 0635 the final step in the hypothesis testing procedure is to come to a final decision regarding the null hypothesis. this is accomplished by comparing the p - value with the level of significance. in this example, the p - value is 0. 127 and the level of significance is 0. 10 : since the p - value level of signific... | openstax_principles-of-data-science-web | [
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as part of their coursework is less than 45 %. to test the claim, the professor selects a sample of 200 students and surveys the students to determine if they use ai tools as part of their coursework. the results from the survey indicate that 74 out of 200 students use ai tools. use this sample data to test the claim m... | openstax_principles-of-data-science-web | [
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of the test statistic. the area under the normal curve to the left of a z - score of is approximately 0. 012. thus, the p - value for this hypothesis test is 0. 012. here is the python code to calculate the p - value for this example : # from scipy. stats import norm from scipy. stats import norm # define parameters x,... | openstax_principles-of-data-science-web | [
0.0257804486900568,
0.02258545346558094,
0.05240704491734505,
-0.022266393527388573,
-0.07173338532447815,
-0.021656109020113945,
0.004409382585436106,
0.0038798819296061993,
-0.052036743611097336,
0.08239845931529999,
0.021327607333660126,
-0.011065380647778511,
0.03136690706014633,
0.009... |
0. 0115 the final step in the hypothesis testing procedure is to come to a final decision regarding the null hypothesis. this is accomplished by comparing the p - value with the level of significance. in this example, the p - value is 0. 012 and the level of significance is 0. 05 : since the p - value level of signific... | openstax_principles-of-data-science-web | [
0.04462868347764015,
0.007156461011618376,
0.018111197277903557,
-0.010326589457690716,
-0.04266371577978134,
-0.006917753256857395,
-0.0010687510948628187,
0.015905948355793953,
-0.004400428384542465,
0.0882086306810379,
0.006838372442871332,
-0.000988319399766624,
0.007004091516137123,
-... |
complement of a “ greater than ” symbol is a “ less than or equals ” symbol, which will be used in the null hypothesis. the claim refers to the population mean oxygen level, so the symbol will be used. the claim corresponds to the alternative hypothesis. notice that this setup will correspond to setup b from table 4. 6... | openstax_principles-of-data-science-web | [
0.024541782215237617,
0.009730213321745396,
0.015111255459487438,
-0.008894704282283783,
-0.02679263800382614,
-0.02044980227947235,
0.03244984149932861,
0.02037697657942772,
-0.014620056375861168,
0.06212032213807106,
0.021428002044558525,
0.00878151599317789,
0.012689772993326187,
-0.011... |
", one _ tailed _ p _ value ) the resulting output will look like this : test statistic = 0. 6512171447631733 p - value = 0. 2655900985964262 the final step in the hypothesis testing procedure is to come to a final decision regarding the null hypothesis. this is accomplished by comparing the p - value with the level of... | openstax_principles-of-data-science-web | [
0.05114743858575821,
0.015309929847717285,
0.03775576502084732,
-0.022916710004210472,
-0.028306173160672188,
-0.02180788293480873,
-0.012428260408341885,
0.006798080634325743,
-0.01805494911968708,
0.06962818652391434,
-0.005193013697862625,
0.0012481254525482655,
-0.00528786750510335,
0.... |
4. 2 • hypothesis testing 179 conclusion : the decision is to fail to reject the null hypothesis ; the claim corresponds to the alternative hypothesis. this scenario corresponds to row 4 in table 4. 8, which means “ there is not enough evidence to support the claim ” that the average oxygen level for female patients is... | openstax_principles-of-data-science-web | [
0.018917731940746307,
0.018483171239495277,
-0.01090454775840044,
-0.01458799745887518,
-0.03719416260719299,
0.0056215329095721245,
-0.006500363349914551,
0.012838620692491531,
0.036870673298835754,
0.09596747159957886,
0.01853697933256626,
-0.028334612026810646,
0.014899292029440403,
-0.... |
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