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Chapter 4: Problem 150

In Exercises 4.150 to \(4.152,\) a confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(95 \%\) confidence interval for \(p: 0.48\) to 0.57 (a) \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (b) \(H_{0}: p=0.75\) vs \(H_{a}: p \neq 0.75\) (c) \(H_{0}: p=0.4\) vs \(H_{a}: p \neq 0.4\)

Short Answer

Expert verified
For hypothesis (a), we do not reject the null hypothesis. For hypotheses (b) and (c), we reject the null hypothesis.

Step by step solution

01

Understanding confidence intervals in hypothesis testing

When a confidence interval is given, any value within the interval can be considered plausible for the population parameter. Conversely, any value outside this interval would be considered implausible and we would reject the null hypothesis if it asserts such a value.
02

Testing hypothesis (a)

In this step, the null hypothesis \(H_{0}: p=0.5\) and the alternative hypothesis \(H_{a}: p \neq 0.5\) are considered. The value \(p=0.5\) is inside the 95% confidence interval given (0.48 to 0.57), hence, we do not reject the null hypothesis.
03

Testing hypothesis (b)

For the null hypothesis \(H_{0}: p=0.75\) and the alternative hypothesis \(H_{a}: p \neq 0.75\), the value \(p=0.75\) is outside the 95% confidence interval (0.48 to 0.57). Therefore, we reject the null hypothesis.
04

Testing hypothesis (c)

The final null hypothesis \(H_{0}: p=0.4\) and alternative hypothesis \(H_{a}: p \neq 0.4\) are evaluated. The value \(p=0.4\) is outside the 95% confidence interval (0.48 to 0.57), hence, we reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
Understanding what a confidence interval represents is key to interpreting the results of many statistical analyses. A confidence interval gives us a range of values, calculated from the data sample, that is likely to contain the population parameter of interest. For example, a 95% confidence interval for a population proportion indicates that we can be 95% confident that the actual proportion lies within the given range.

Null Hypothesis
At the heart of hypothesis testing is the null hypothesis, often denoted as \(H_0\). It is a statement or assumption that there is no effect or no difference, and it serves as a starting point for testing. In the context of our exercise, the null hypotheses, like \(H_0: p=0.5\), represents a claim about the population proportion which we are testing against the sample data. If the evidence does not strongly contradict it, we refrain from rejecting the null.

Alternative Hypothesis
The alternative hypothesis, denoted as \(H_a\) or \(H_1\), is the statement that directly challenges the null hypothesis. It often represents the assumption that there is an effect, or there is a difference. For instance, when we say \(H_a: p eq 0.5\), we're asserting that the true population proportion is not 0.5. Where the null hypothesis is conservatively cautious, the alternative hypothesis is more of a claim needing substantial evidence to be upheld.

Significance Level
The significance level is a threshold of probability set before testing the hypothesis and is denoted by alpha (\(\alpha\)). It represents the maximum probability that we are willing to accept for wrongly rejecting the null hypothesis, termed as a Type I error. Common levels include 0.05 (5%), 0.01 (1%), and 0.10 (10%). In our exercise, we implicitly used a significance level of 5% since we were dealing with a 95% confidence interval; this corresponds to a 5% risk of concluding that there is an effect when there is none.

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Most popular questions from this chapter

Beer and Mosquitoes Does consuming beer attract mosquitoes? A study done in Burkino Faso, Africa, about the spread of malaria investigated the connection between beer consumption and mosquito attraction. \(^{9}\) In the experiment, 25 volunteers consumed a liter of beer while 18 volunteers consumed a liter of water. The volunteers \({ }^{8}\) Bouchard, M., Bellinger, D., Wright, \(\mathrm{R}\), and Weisskopf, M. "Attention-Deficit/Hyperactivity Disorder and Urinary Metabolites of Organophosphate Pesticides," Pediatrics, \(2010 ; 125:\) e1270-e1277. \({ }^{9}\) Lefvre, T., et al., "Beer Consumption Increases Human Attractiveness to Malaria Mosquitoes," PLoS ONE, 2010; 5(3): e9546.were assigned to the two groups randomly. The attractiveness to mosquitoes of each volunteer was tested twice: before the beer or water and after. Mosquitoes were released and caught in traps as they approached the volunteers. For the beer group, the total number of mosquitoes caught in the traps before consumption was 434 and the total was 590 after consumption. For the water group, the total was 337 before and 345 after. (a) Define the relevant parameter(s) and state the null and alternative hypotheses for a test to see if, after consumption, the average number of mosquitoes is higher for the volunteers who drank beer. (b) Compute the average number of mosquitoes per volunteer before consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (c) Compute the average number of mosquitoes per volunteer after consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (d) If the difference in part (c) is unlikely to happen by random chance, what can we conclude about beer consumption and mosquitoes? (e) If the difference in part (c) is statistically significant, do we have evidence that beer consumption increases mosquito attraction? Why or why not?

For each situation described in Exercises 4.93 to 4.98 , indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level \((\) such as \(\alpha=0.01)\) A pharmaceutical company is testing to see whether its new drug is significantly better than the existing drug on the market. It is more expensive than the existing drug. Which significance level would the company prefer? Which significance level would the consumer prefer?

Influencing Voters: Is a Phone Call More Effective? Suppose, as in Exercise \(4.37,\) that we wish to compare methods of influencing voters to support a particular candidate, but in this case we are specifically interested in testing whether a phone call is more effective than a flyer. Suppose also that our random sample consists of only 200 voters, with 100 chosen at random to get the flyer and the rest getting a phone call. (a) State the null and alternative hypotheses in this situation. (b) Display in a two-way table possible sample results that would offer clear evidence that the phone call is more effective. (c) Display in a two-way table possible sample results that offer no evidence at all that the phone call is more effective. (d) Display in a two-way table possible sample results for which the outcome is not clear: There is some evidence in the sample that the phone call is more effective but it is possibly only due to random chance and likely not strong enough to generalize to the population.

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: \mu=10\) vs \(H_{a}: \mu>10\) Sample: \(\bar{x}=12, s=3.8, n=40\) Randomization statistic \(=\bar{x}\)

Testing for a Gender Difference in Compassionate Rats In Exercise 3.80 on page 203 , we found a \(95 \%\) confidence interval for the difference in proportion of rats showing compassion, using the proportion of female rats minus the proportion of male rats, to be 0.104 to \(0.480 .\) In testing whether there is a difference in these two proportions: (a) What are the null and alternative hypotheses? (b) Using the confidence interval, what is the conclusion of the test? Include an indication of the significance level. (c) Based on this study would you say that female rats or male rats are more likely to show compassion (or are the results inconclusive)?
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