10 Jan 29 introduction to hypothesis testing

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University Of Georgia *

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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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