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Chapter 8: Problem 61

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.

Short Answer

Expert verified
When comparing two sample proportions with a two-sided alternative hypothesis, the p-value will be smaller if the sample proportions are far apart. This is because far-apart proportions indicate that the observable difference is less likely due to chance.

Step by step solution

01

Understanding p-value

The p-value is a crucial measurement in hypothesis testing. If it is small (typically 鈮� 0.05), the null hypothesis is rejected in favor of the alternative hypothesis. Conversely, a larger p-value suggests that changes in your observations are likely due to chance and not due to the variables you're measuring.
02

Evaluating p-value in two sample proportion test.

In a two-sample proportion test, the p-value helps understand whether the difference in the proportion of successes in two groups could have happened by chance. If the sample proportions are close to each other, then the differences observed could more likely be due to random chance, hence a larger p-value. On the other hand, if the sample proportions are far apart, it is less likely that the difference is due to chance, hence a smaller p-value.
03

Conclusion

Therefore, when comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, one would get a smaller p-value if the sample proportions are far apart. This is because when sample proportions are far apart, it signifies that the difference between the groups is less likely to be due to chance, thereby leading to a smaller p-value.

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Most popular questions from this chapter

A researcher carried out a hypothesis test using a two-sided alternative hypothesis. Which of the following \(z\) -scores is associated with the smallest p-value? Explain. $$ \text { i. } z=0.50 $$ ii. \(z=1.00\) iii. \(z=2.00\) iv. \(z=3.00\)

A coin is flipped 30 times and comes up heads 18 times. You want to test the hypothesis that the coin does not come up \(50 \%\) heads in the long run. Pick the correct null hypothesis for this test. i. \(\mathrm{H}_{0}: \hat{p}=0.60\) ii. \(\mathrm{H}_{0}: p=0.50\) iii. \(\mathrm{H}_{0}: p=0.60\) iv. \(\mathrm{H}_{0}=\hat{p}=0.50\)

Anxiety Disorders According to Time magazine (December 5,2011 ), \(18 \%\) of adult Americans suffer from an anxiety disorder. Suppose a test of 300 random college students showed that 50 suffered from an anxiety disorder a. How many out of 300 would you expect to have an anxiety disorder if the \(18 \%\) is correct? b. Suppose you are testing the hypothesis that the population proportion of college students suffering from an anxiety disorder is not \(0.18\) at the \(0.05\) significance level. Choose the correct figure and interpret the p-value.

Choose one of the answers in each case. In statistical inference, measurements are made on a (sample or population), and generalizations are made to a (sample or population).

Mercury in Freshwater Fish (Example 11) Some experts believe that \(20 \%\) of all freshwater fish in the United States have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 250 fish tested, and 60 of them have dangerous levels of mercury. Test the hypothesis that this sample is not from a population with \(20 \%\) dangerous fish. Use a significance level of \(0.05\). Comment on your conclusion: Are you saying that the percentage of dangerous fish is definitely \(20 \%\) ? Explain.
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