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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
In the comparison of two sample proportions with a two-sided alternative hypothesis, you will get a smaller p-value with a larger sample size, assuming that all other factors remain equal. This is because a larger sample size yields more precise estimates, which results in a larger test statistic and a smaller p-value.

Step by step solution

01

Understanding the P-Value

The p-value is a statistical measure that helps us to decide whether to reject the null hypothesis. It measures the strength of evidence in support of a null hypothesis, which is the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error. It's generally understood that a smaller p-value indicates strong evidence against the null hypothesis, while a larger p-value would suggest there's not strong enough evidence to reject it.
02

Relation Between P-Value and Sample Size

Generally, as the sample size increases, the p-value will decrease, assuming all other factors remain constant. This is because a larger sample size gives us more precise estimates of population parameters, thus reducing standard errors, which in turn makes the test statistic larger. Since the p-value is determined by the test statistic, a larger test statistic results in a smaller p-value.
03

Effect of Sample Size on P-Value - Example

For example, let's say we have two studies: one with a sample size of 500 and another with a sample size of 5000. Suppose both studies achieve the same result in terms of difference between sample proportions. Although the observed differences are the same, the one with a larger sample size (5000) would yield a smaller p-value, assuming that all other factors are equal. This is due to the increased precision of the estimates gained by using a larger sample size, which in turn leads to a larger test statistic and thus smaller p-value.

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Most popular questions from this chapter

According to a Gallup poll, \(11.55 \%\) of American adults have diabetes. Suppose a researcher wonders if the diabetes rate in her area is higher than the national rate. She surveys 150 adults in her area and finds that 21 of them have diabetes. a. If the region had the same rate of diabetes as the rest of the country, how many would we expect have diabetes? b. Suppose you are testing the hypothesis that the diabetes rate in this area differs from the national rate, using a \(0.05\) significance level. Choose the correct figure and interpret the p-value.

A community college used enrollment records of all students and reported that that the percentage of the student population identifying as female in 2010 was \(54 \%\) whereas the proportion identifying as female in 2018 was \(52 \%\). Would it be appropriate to use this information for a hypothesis test to determine if the proportion of students identifying as female at this college had declined? Explain.

Morse determined that the percentage of \(t\) 's in the English language in the 1800 s was \(9 \%\). A random sample of 600 letters from a current newspaper contained \(48 t\) 's. Using the \(0.10\) level of significance, test the hypothesis that the proportion of \(t\) 's in this modern newspaper is \(0.09\).

Give the null and alternative hypotheses for each test, and state whether a one-proportion z-test or a two-proportion z-test would be appropriate. a. You test a person to see whether he can tell tap water from bottled water. You give him 20 sips selected randomly (half from tap water and half from bottled water) and record the proportion he gets correct to test the hypothesis. b. You test a random sample of students at your college who stand on one foot with their eyes closed and determine who can stand for at least 10 seconds, comparing athletes and nonathletes.

In each case. choose whether the appropriate test is a one-proportion \(z\) -test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.
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