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Chapter 4: Problem 157

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu=15\) vs \(H_{a}: \mu \neq 15\) (a) \(95 \%\) confidence interval for \(\mu: \quad 13.9\) to 16.2 (b) \(95 \%\) confidence interval for \(\mu: \quad 12.7\) to 14.8 (c) \(90 \%\) confidence interval for \(\mu: \quad 13.5\) to 16.5

Short Answer

Expert verified
For samples (a) and (c), we do not reject the null hypothesis at 5% and 10% significance levels respectively while for sample (b), we reject the null hypothesis at a 5% significance level.

Step by step solution

01

Analyze the first confidence interval

For part (a), the confidence interval ranges from 13.9 to 16.2. Since the hypothesized population mean of 15 lies within this interval, we do not reject the null hypothesis at a 5% significance level (we are using a 95% confidence interval).
02

Analyze the second confidence interval

For part (b), the confidence interval ranges from 12.7 to 14.8. The hypothesized population mean of 15 does not fall within this interval, so we reject the null hypothesis at a 5% significance level.
03

Analyze the third confidence interval

For part (c), the confidence interval ranges from 13.5 to 16.5. Since the hypothesized population mean of 15 is within this range, we do not reject the null hypothesis at a 10% significance level (because we are using a 90% confidence interval).

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values, derived from the sample data, that is likely to contain the population parameter of interest. When we calculate, for example, a 95% confidence interval, we are saying that we are 95% confident that the true population mean falls within this range.

In the context of the exercise, the confidence intervals are given for the population mean \( \mu \). When the sample mean falls within the confidence interval, such as in parts (a) and (c), we do not have sufficient evidence to reject the null hypothesis. This is because the hypothesized value of \( \mu=15 \) is within the range that we're 95% or 90% confident contains the true mean.

However, for part (b), the hypothesized mean does not fall within the 95% confidence interval. This suggests that the true mean is likely different from 15, and we would reject the null hypothesis in this scenario. Confidence intervals are an essential part of statistical hypothesis testing because they provide a range of plausible values for the parameter, allowing for a decision on the null hypothesis based on the data at hand.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference, and it generally represents a skeptical perspective or a claim to be tested. In hypothesis testing, we seek to determine whether the evidence suggests that we should reject this null hypothesis in favor of an alternative hypothesis, denoted as \( H_a \).

For instance, in the given exercise, \( H_0: \mu=15 \) asserts that the population mean is 15. The alternative hypothesis \( H_a: \mu eq 15 \) posits that the population mean is not 15. The null hypothesis is the starting assumption for the test, and the hypothesis testing procedure examines whether the data collected provides enough evidence to conclude if the null hypothesis can be rejected or not.

As we can see from the solutions, a confidence interval that does not include the value stated in the null hypothesis (as in part (b)) is an indicator that the null hypothesis may not hold. Conversely, when the confidence interval includes the null hypothesis value, we lack evidence to reject it (as seen in parts (a) and (c)).
Significance Level
The significance level, often denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. It represents the researcher's tolerance for such errors and is a critical value in hypothesis testing that helps determine the threshold for rejecting the null hypothesis.

Common significance levels are 5% (0.05), 1% (0.01), or 10% (0.10), which corresponds inversely to 95%, 99%, and 90% confidence levels, respectively. Demonstrated in our exercise, when we reject the null hypothesis for part (b), it's because the 95% confidence interval does not include the hypothesized mean of 15, thus surpassing the 5% significance level criterion for rejection.

Alternatively, for parts (a) and (c), the hypothesized mean lies within the confidence intervals, indicating that we do not have significant evidence at the 5% and 10% levels, respectively, to reject the null hypothesis. Deciding on the appropriate significance level is a crucial step in the design of an experiment or study as it can influence the conclusions drawn from the statistical test.

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Most popular questions from this chapter

Could owning a cat as a child be related to mental illness later in life? Toxoplasmosis is a disease transmitted primarily through contact with cat feces, and has recently been linked with schizophrenia and other mental illnesses. Also, people infected with Toxoplasmosis tend to like cats more and are 2.5 times more likely to get in a car accident, due to delayed reaction times. The CDC estimates that about \(22.5 \%\) of Americans are infected with Toxoplasmosis (most have no symptoms), and this prevalence can be as high as \(95 \%\) in other parts of the world. A study \(^{37}\) randomly selected 262 people registered with the National Alliance for the Mentally Ill (NAMI), almost all of whom had schizophrenia, and for each person selected, chose two people from families without mental illness who were the same age, sex, and socioeconomic status as the person selected from NAMI. Each participant was asked whether or not they owned a cat as a child. The results showed that 136 of the 262 people in the mentally ill group had owned a cat, while 220 of the 522 people in the not mentally ill group had owned a cat. (a) This is known as a case-control study, where cases are selected as people with a specific disease or trait, and controls are chosen to be people without the disease or trait being studied. Both cases and controls are then asked about some variable from their past being studied as a potential risk factor. This is particularly useful for studying rare diseases (such as schizophrenia), because the design ensures a sufficient sample size of people with the disease. Can casecontrol studies such as this be used to infer a causal relationship between the hypothesized risk factor (e.g., cat ownership) and the disease (e.g., schizophrenia)? Why or why not? (b) In case-control studies, controls are usually chosen to be similar to the cases. For example, in this study each control was chosen to be the same age, sex, and socioeconomic status as the corresponding case. Why choose controls who are similar to the cases? (c) For this study, calculate the relevant difference in proportions; proportion of cases (those with schizophrenia) who owned a cat as a child minus proportion of controls (no mental illness) who owned a cat as a child. (d) For testing the hypothesis that the proportion of cat owners is higher in the schizophrenic group than the control group, use technology to generate a randomization distribution and calculate the p-value. (e) Do you think this provides evidence that there is an association between owning a cat as a child and developing schizophrenia? \(^{38}\) Why or why not?

Match the four \(\mathrm{p}\) -values with the appropriate conclusion: (a) The evidence against the null hypothesis is significant, but only at the \(10 \%\) level. (b) The evidence against the null and in favor of the alternative is very strong. (c) There is not enough evidence to reject the null hypothesis, even at the \(10 \%\) level. (d) The result is significant at a \(5 \%\) level but not at a \(1 \%\) level. I. 0.00008 II. 0.0571 III. 0.0368 IV. \(\quad 0.1753\)

For each situation described, 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 \(\alpha=0.01\) ). Testing to see if a well-known company is lying in its advertising. If there is evidence that the company is lying, the Federal Trade Commission will file a lawsuit against them.

For each situation described, 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 \(\alpha=0.01\) ). Using a sample of 10 games each to see if your average score at Wii bowling is significantly more than your friend's average score.

The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?
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