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Chapter 8: Problem 62

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.

Short Answer

Expert verified
Given all other factors are equal, a larger sample size will generally lead to a smaller p-value during the comparison of two sample proportions with a two-sided alternative hypothesis. This is because the larger the sample size, the better the estimation it gives of the population parameters.

Step by step solution

01

Understanding the Concept of P-value

The p-value is a statistical measure used in hypothesis testing to decide whether to reject or accept the null hypothesis. It indicates the probability of observing the data given that the null hypothesis is true.
02

Effect of sample size on the p-value

Increasing the sample size generally allows for better estimation of the population parameters. The effect of sample size on p-value depends on the variation in the population. If there is a large variation, a larger sample size will provide a more accurate estimate of the population parameter, leading to a smaller p-value. So, in comparison, a larger sample size will generally lead to a smaller p-value, all other factors being equal.
03

Application in Two-sided alternative hypothesis

In a two-sided alternative hypothesis test, we are testing for a difference in any direction from the hypothesized value. Therefore, a larger sample size gives us more power to detect a difference, if one exists, and hence is more likely to lead to a smaller p-value.

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Most popular questions from this chapter

By establishing a small value for the significance level, are we guarding against the first type of error (rejecting the null hypothesis when it is true) or guarding against the second type of error?

Give the null and alternative hypothesis for each test, and state whether a oneproportion z-test or a two-proportion z-test would be appropriate. a. You test a random sample of eighth-grade students who play daily for 3 hours and devote the same time to homework, comparing boys and girls. b. You test a person to see whether she can tell butter on bread from margarine spread on bread. You give her 20 toast bits selected randomly (half with butter and half with margarine) and record the proportion she gets correct to test the hypothesis.

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Suppose you wanted to test the claim that more than half of French citizens support revising the tax regime and structure. Give the null and alternative hypotheses, and explain, using both words and symbols.

Historically (from about 2001 to 2005 ), about \(58 \%\) of Americans believed that Earth's temperature was rising ("global warming"). A March 2010 Gallup Poll sought to determine whether this proportion had changed. The poll interviewed 1014 adult Americans, and 527 said they believed that global warming was real. (Assume these 1014 adults represented a simple random sample.) a. What percentage in the sample believed global warming was real in 2010 ? Is this more or less than the historical \(58 \%\) ? b. Test the hypothesis that the proportion of Americans who believe global warming is real has changed. Use a significance level of \(0.05\). c. Choose the correct interpretation: j. In 2010 , the percentage of Americans who believe global warming is real is not significantly different from \(58 \%\). ii. In 2010 , the percentage of Americans who believe global warming is real has changed from the historical level of \(58 \%\).
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