Interesting Results Suppose you wish to find out the answer to the age-old question, "Do Americans prefer Coke or Pepsi?" You conduct a blind taste test in which individuals are randomly asked to drink one of the colas first, followed by the other cola, and then asked to disclose which drink they prefer. Results of your taste test indicate that 53 of 100 individuals prefer Pepsi.
c. Suppose you changed the level of significance in conducting the hypothesis test to α = 0.01. What would happen to the range of values of p₀ for which the null hypothesis is not rejected? Why does this make sense?

Verified step by step guidance

1. Recall that the level of significance \( \alpha \) represents the probability of rejecting the null hypothesis when it is actually true (Type I error). When you decrease \( \alpha \) from 0.05 to 0.01, you are making the test more stringent.
2. Understand that the range of values of \( p_0 \) for which the null hypothesis is not rejected corresponds to the confidence interval around the sample proportion. A smaller \( \alpha \) means a higher confidence level (e.g., 99% instead of 95%), which results in a wider confidence interval.
3. Because the confidence interval becomes wider, the range of \( p_0 \) values for which you fail to reject the null hypothesis increases. This means you need stronger evidence from the sample to reject the null hypothesis at the 0.01 level compared to the 0.05 level.
4. Mathematically, the critical value \( z_{\alpha/2} \) increases as \( \alpha \) decreases, which affects the margin of error in the confidence interval formula: \[ \text{Margin of Error} = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.
5. In summary, lowering \( \alpha \) to 0.01 makes the test more conservative, increasing the range of \( p_0 \) values where the null hypothesis is not rejected, which makes sense because you are reducing the chance of a false positive and thus require stronger evidence to reject the null.

Key Concepts

Level of Significance (α)

The level of significance, denoted by α, is the threshold probability for rejecting the null hypothesis. It represents the maximum risk of a Type I error (false positive) that a researcher is willing to accept. Lowering α (e.g., from 0.05 to 0.01) makes the test more stringent, requiring stronger evidence to reject the null hypothesis.

Null Hypothesis and Rejection Region

The null hypothesis (H₀) is a statement of no effect or no difference, tested against an alternative hypothesis. The rejection region consists of values of the test statistic that lead to rejecting H₀. Changing α alters the size of this region: a smaller α shrinks the rejection region, making it harder to reject H₀.

Confidence Intervals and Hypothesis Testing Relationship

The range of p₀ values for which the null hypothesis is not rejected corresponds to the confidence interval for the population parameter. Decreasing α widens the confidence interval, meaning more values of p₀ are considered plausible, which aligns with requiring stronger evidence to reject H₀.