Hypothesis Testing Using Rejection Regions In Exercises 7–12, ...

Question

Hypothesis Testing Using Rejection Regions In Exercises 7–12, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Vaccinations In 2021, a reporter claims that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. In a random sample of 200 U.S. adults, 56% feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. At α=0.10, is there enough evidence to reject the reporter’s claim?

Accepted Answer

All right, hello, everyone. So this question says, a survey analyst claims that at least 60% of consumers support stricter labeling laws for organic products. A recent random sample of 180 consumers shows that 102 of them support stricter labeling laws. Using a significance level of alpha equals 0.10, is there enough evidence to reject the analyst's claim? All right, so first, let's define the hypothesis for this survey, for this experiment. Now, the claim is that the P value should be greater than or equal to 60% or 0.60%. Because we want to know if we can reject that claim. This is going to correspond to the null hypothesis. So, the null hypothesis states that P is greater than or equal to 0.60. By contrast, the alternative hypothesis would state instead that P is less than 0.60. And this is a left tailed test. So we already know the significance level, right? It's already established that alpha is equal to 0.10. So, using this information for a left-tailed test, the critical value, according to our Z table, is equal to -1.28, according to the standard normal distribution. And now we can find our rejection region. So, we can reject the null hypothesis, we can reject the claim. If our calculated Z value is less than -1.28. Now, speaking of the test statistic, we would use the Z test for a proportion formula. So Z Our test statistic Is equal to Pihat Subtracted by P0 or P sub zero. And divided by The square root of p knot multiplied by one subtracted by Pan knot. Divided by and Now Phat corresponds to the sample proportion. So here it's the proportion of people in the survey that agreed to supporting stricter labeling laws. So Phat is equal to 102 out of or divided by 180. Which equals approximately 0.5667. So going back to our formula. Z is equal to 0.5667. Subtracted by 0.60. Divided by The square root of. 0.60 multiplied by 1 subtracted by 0.60. And divided by 180. Ultimately, Z is equal to approximately 0.91. Which leads you now to your comparison and then your decision. So here we can see that our calculated test statistic -0.91. Is greater than -1.28. In other words, it is not in the rejection region. Because of this, we ultimately fail to reject. The null hypothesis. So now this leads us to our final answer, which I went ahead and rewrote on the screen here. Our final answer reads as follows. There is not enough evidence to reject the claim that at least 60% support stricter labeling laws. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Hypothesis Testing

Rejection Region

Standardized Test Statistic

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