10 Jan 29 introduction to hypothesis testing

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Apr 3, 2024

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Introduction to Hypothesis Testing Example: Side Effects According to a website, 20% experience dizziness as a result of using a particular drug. The pharmaceutical company has made improvements to the drug and hopes to reduce this percentage. What is the variable? Is it categorical or quantitative? Categorical; whether nor not someone experiences dizziness using the new drug. Define the parameter of interest in the context of the problem. p = proportion or probability someone experiences dizziness using the new drug. How should we define a success and a failure? Success = dizziness, Failure = no dizziness Claim : The proportion of all people that experience dizziness when using the new drug is still 0.2. Question : Is the proportion that experience dizziness when using the new drug now less than 0.2? The null hypothesis is a statement we assume to be true. The alternative hypothesis is the statement that we want to determine if there is evidence to support. Determine the null and alternative hypotheses. Null Hypothesis (H0): p = 0.2 Alternative Hypothesis (Ha): p < 0.2 Use parameters, not statistics; Ha can be <, >, ≠ (assuming H0 is true, so Ha can’t be) 1

Since the alternative hypothesis is Ha: p<0.2, sample proportions in the left tail would be of interest. Below is a simulation of 10,000 sample proportions assuming the true proportion is 0.2. To determine if there is enough evidence to conclude the proportion of people that experience dizziness has been reduced, we need to find the probability that p ˆ is less than or equal to the one obtained. Things that occur in the middle are random, occurring by chance. In the following table, there are various observed sample proportions to consider. For each, find the probability of obtaining a sample proportion as extreme or more extreme than the one obtained. Observ ed p ˆ Probability 0.175 (2+28+65+218+535+779+1259+1560) /10000 = 0.4446 0.1 (2+28+65+218+535)/10000 = 0.0848 0.05 0.0095 2

These probabilities are referred to as p-values . Sample proportions near the hypothesized value, 0.2, could happen by chance alone and correspond to a high probability. But, sample proportions far from the hypothesized value could not have happened by chance alone and result in smaller probabilities. But, how small? This could be decided by using a cutoff which is referred to as the level of significance . Common values are 0.01, 0.05 or 0.1. The level of significance should be decided before collecting any data. If the p-value is less than or equal to the level of significance, • the decision will be: reject the null hypothesis • the conclusion will be: there is enough evidence to conclude H a . If the p-value is greater than the level of significance, • the decision will be: fail to reject the null hypothesis • the conclusion will be: there is not enough evidence to conclude H a . Never state the null hypothesis in the conclusion. There is not enough evidence to conclude the probability someone experiences dizziness using the new drug is less than 0.2. 3

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Example: Reincarnation According to a poll conducted in the 1970s, 9% believe in reincarnation. Suppose we want to determine if this percentage has changed. To test this, a sample of 480 adults finds that 53 believe in reincarnation. Assume that the sample is representative of all adults in the population. What is the variable? Is it categorical or quantitative? Categorical; whether or not someone currently believes in reincarnation. Define the parameter in the context of the problem. p = the proportion of people that currently believe in reincarnation Determine the null and alternative hypotheses. H0: There is no difference in the percentage of adults that believe in reincarnation (p = 0.09). Ha: The percentage of adults that believe in incarnation is greater than or less than 0.09 (p ≠ 0.09). Conditions: • Why can we conclude the responses of the 480 people in the sample can be regarded as independent? Representative sample implies there must have been some random sampling. • Why can we conclude the sampling distribution of p ˆ is nearly normal? np = 480*0.09 = 43.2 n(1-p) = 480*0.91) = 436.8 Because both values are greater than 15, we can conclude that the sampling distribution of ^ p is nearly normal. Considering the alternative hypothesis, what values of p ˆ will make you think we should reject the claim in favor of the alternative? a) any sample proportion that does not equal 0.09 b) any sample proportion that is either too small or too large 4

c) only sample proportions that are too small d) only sample proportions that are too large Assume the null hypothesis is true. Determine the mean and standard deviation of the sampling distribution of p ˆ. Then, find the z-score for the observed sample proportion. Mean = 0.09 SD = √ p ( 1 − p ) n =0.0131 Observed ^ p = 53/480 = 0.1104 Z-Score = 0.1104 − 0.09 0.0131 = 1.557 (test statistic) Since the sampling distribution is approximately normal, we can use the standard normal distribution to find the probability of obtaining a test statistic as extreme or more extreme than the one obtained. (Mean = 0, SD = 1) test statistic Use JMP Normal Distribution Calculator 5 −3 −2 −1 0 1 2 3

p = 0.1195 At the 0.05 level of significance, what is the decision: reject or fail to reject the null hypothesis? Fail to reject State the conclusion in the context of the problem. There is not enough evidence to conclude that the proportion of people that currently believe in reincarnation is different from (not equal to) 0.09. Have we proven the current percentage that believe in reincarnation is still 9%? *Never want to state the null hypothesis in the conclusion; never say “accept H 0 ”. 6

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, Steps in a Hypothesis Test 1. Describe the population characteristic about which hypotheses are to be tested. 2. State the null and alternative hypotheses. 3. Select the level of significance, α . 4. Check to make sure any conditions required for the test are reasonable. 5. Compute the p-value. 6. State the decision: reject H 0 or fail to reject H 0 . Then, state the conclusion in the context of the problem. Example: Dress for Success In a survey conducted by CareerBuilder.com, employers were asked if they had ever sent an employee home because they were dressed inappropriately. A total of 2,765 employers were randomly selected, with 968 saying that they had sent an employee home for inappropriate attire. Do the sample data provide convincing evidence that more than one- third of employers have sent an employee home to change clothes? Use a level of significance of 0.05. Variable: Categorical; whether or not they have sent someone home for inappropriate attire. p = proportion of employers that have sent an employee home for inappropriate attire. H0: p = 1/3 Ha: p > 1/3 Conditions: 2765 employers were randomly selected p-value = P(968 or more when p = 1/3 successes) Don’t have to check normality when using binomial or simulation (only normal). 7

Since 0.0323 <= 0.05, reject H0. There is enough evidence to conclude the proportion of employers that have sent an employee home for inappropriate attire is more than 1/3. 8

10 Jan 29 introduction to hypothesis testing

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