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Chapter 17: Problem 43

Family caregivers of patients with chronic illness can experience anxiety. Do regular support-group meetings affect these feelings of anxiety? It is possible that they reduce anxiety, perhaps through sharing experiences with other caregivers in similar situations, or increase anxiety, perhaps by reinforcing painful experiences by recounting them to others. To explore the effect of support-group meetings, several familiy caregivers were enrolled in a support group. After three months, researchers administered a test to measure anxiety, with larger scores indicating greater anxiety. Assume these caregivers are a random sample from the population of all family caregivers. A \(95 \%\) confidence interval for the population mean anxiety score \(\mu\) after participating in a support group is \(7.2 \pm 0.7 .{ }^{13}\) Use the method described in the previous exercise to answer these questions. (a) Suppose we know that the mean anxiety score for the population of all family caregivers is \(6.4\). With a two-sided alternative, can you reject the null hypothesis that \(\mu=6.4\) at the \(5 \%(\alpha=0.05)\) significance level? Why? (b) Suppose we know that the mean anxiety score for the population of all family caregivers is \(6.6\). With a two-sided alternative, can you reject the null hypothesis that \(\mu=6.6\) at the \(5 \%(\alpha=0.05)\) significance level? Why?

Short Answer

Expert verified
(a) Reject null hypothesis; 6.4 is outside CI. (b) Do not reject null hypothesis; 6.6 is within CI.

Step by step solution

01

Understand Confidence Interval

The confidence interval given is \(7.2 \pm 0.7\). This means the interval is \([6.5, 7.9]\). A confidence interval is a range of values that is likely to contain the population mean with a certain level of confidence, which in this case is 95%.
02

Formulate Hypotheses for Part (a)

For part (a), we need to test \(H_0: \mu = 6.4\) against \(H_a: \mu eq 6.4\). We see if \(6.4\) lies within the confidence interval \([6.5, 7.9]\).
03

Evaluate Hypothesis for Part (a)

The value \(6.4\) is not within the confidence interval \([6.5, 7.9]\). Therefore, we can reject the null hypothesis \(H_0\) at the 5% significance level.
04

Formulate Hypotheses for Part (b)

For part (b), we need to test \(H_0: \mu = 6.6\) against \(H_a: \mu eq 6.6\). We see if \(6.6\) lies within the confidence interval \([6.5, 7.9]\).
05

Evaluate Hypothesis for Part (b)

The value \(6.6\) is within the confidence interval \([6.5, 7.9]\). Therefore, we cannot reject the null hypothesis \(H_0\) at the 5% significance level.

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Key Concepts

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

Confidence Interval
When researchers talk about a confidence interval, they are referring to a range of values within which we can say, with a certain level of confidence, that the true population parameter lies. In our exercise, the confidence interval is expressed as \(7.2 \pm 0.7\), which means the interval is from 6.5 to 7.9. This signifies that there is a 95% certainty that the true population mean of the anxiety scores for caregivers is within this range.
  • Confidence Level: Represents the percentage of all possible samples that can be expected to include the true population parameter. Here, it is 95%.
  • Margin of Error: The number 0.7 represents the margin of error, showing how much a sample result can differ from reality.
Understanding confidence intervals helps in making inferences about the broader population without having to observe everyone. It's like saying, "We're pretty sure the truth is somewhere in here," allowing us to make decisions or predictions with a certain level of uncertainty.
Null Hypothesis
The null hypothesis is a fundamental concept in hypothesis testing. It is a statement that we aim to test, and it usually proposes no effect or no difference. In this exercise, the null hypotheses are:
  • For part (a): \(H_0: \mu = 6.4\)
  • For part (b): \(H_0: \mu = 6.6\)
When analyzing data, we look at whether we have enough evidence to reject the null hypothesis. For instance, if the hypothesized mean \(\mu\) lies outside the confidence interval, like the 6.4 in part (a), it shows inconsistency with our data, and thus we reject \(H_0\). If \(\mu\) is inside the interval, as with 6.6 in part (b), this suggests there is not enough evidence to reject \(H_0\).
Understanding the null hypothesis gives structure to statistical testing. It helps researchers identify whether there is enough evidence to support an alternative claim or theory.
Significance Level
The significance level in statistics, denoted as \(\alpha\), is the threshold at which we decide whether to reject the null hypothesis. In our scenario, the significance level is set at 5% (or 0.05). This means that there's a 5% risk of concluding that a difference exists when there actually isn't one.
  • If \(\mu\) lies outside the confidence interval, indicating that true parameter is unlikely when the null is true, we reject the null hypothesis.
  • If \(\mu\) is within the interval, we conclude that there isn't enough evidence to reject the null hypothesis.
Choosing the correct significance level is critical, since a too-high level may lead to false positives (Type I errors), while a too-low level might miss actual effects (Type II errors). In essence, the significance level helps us evaluate the strength of our evidence and make informed decisions based on our hypothesis testing.

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Most popular questions from this chapter

A confidence interval for the population mean \(\mu\) tells us which values of \(\mu\) are plausible (those inside the interval) and which values are not plausible (those outside the interval) at the chosen level of confidence. You can use this idea to carry out a test of any null hypothesis \(H_{0}: \mu=\mu_{0}\) starting with a confidence interval: reject \(H_{0}\) if \(\mu_{0}\) is outside the interval and fail to reject if \(\mu_{0}\) is inside the interval. The alternative hypothesis is always two-sided, \(H_{a}: \mu \neq \mu_{0}\) because the confidence interval extends in both directions from \(\mathrm{x}^{-} \bar{x}\). A \(95 \%\) confidence interval leads to a test at the \(5 \%\) significance level because the interval is wrong \(5 \%\) of the time. In general, confidence level \(C\) leads to a test at significance level \(\alpha=1-C\). (a) In Example 17.7, a medical director found mean blood pressure \(\mathrm{x}^{-} \bar{x}=\) \(128.07\) for an SRS of 72 male executives between the ages of 50 and 59 . The standard deviation of the blood pressures of all males \(50-59\) years of age is \(\sigma=15\). Give a \(90 \%\) confidence interval for the mean blood pressure \(\mu\) of all executives in this age group, assuming the standard deviation is the same as for all males \(50-59\) years of age. (b) The hypothesized value \(\mu_{0}=130\) falls inside this confidence interval. Carry out the \(z\) test for \(H_{0}: \mu=130\) against the two-sided alternative. Show that the test is not statistically significant at the \(10 \%\) level. (c) The hypothesized value \(\mu_{0}=131\) falls outside this confidence interval. Carry out the \(z\) test for \(H_{0}: \mu=131\) against the two-sided alternative. Show that the test is statistically significant at the \(10 \%\) level.

Research suggests that pressure to perform well can reduce performance on exams. Are there effective strategies to deal with pressure? In an experiment, researchers had students take a test on mathematical skills. The same students were asked to take a second test on the same skills, but now each student was paired with a partner and only if both improved their scores would they receive a monetary reward for participating in the experiment. They were also told that their performance would be videotaped and watched by teachers and students. To help them cope with the pressure, 10 minutes before the second exam they were asked to write as candidly as possible about their thoughts and feelings regarding the exam. "Students who expressed their thoughts before the high-pressure test showed a statistically significant \(5 \%\) math accuracy improvement from the pretest to posttest" \((P<0.03)\). \(^{9} \mathrm{~A}\) colleague who knows no statistics says that an increase of \(5 \%\) isn't a lotmaybe it's just an accident due to natural variation among the students. Explain in simple language how " \(P<0.03\) " answers this objection.

A randomized comparative experiment examined the effect of the attractiveness of an instructor on the performance of students on a quiz given by the instructor. The researchers found a statistically significant difference in quiz scores between students in a class with an instructor rated as attractive and students in a class with an instructor rated as unattractive \((P=\) \(0.005) \cdot{ }^{7}\) When asked to explain the meaning of " \(P=0.005\)," a student says, "This means there is only probability of \(0.005\) that the null hypothesis is true." Explain what \(P=0.005\) really means in a way that makes it clear that the student's explanation is wrong.

In the study described in Exercise 17.33, researchers also examined the effect of the sex of an instructor on performance of students on a quiz. The researchers found no evidence of a difference in scores. \((P=0.24)\). The \(P\)-value refers to a null hypothesis of "no difference" in quiz scores measured on classes taught by male and female instructors. Explain clearly why this value provides no evidence of a difference.

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus. The sample mean is \(\mathrm{x}^{-} \bar{x}=16.5\) seconds. The null hypothesis for the test of significance is (a) \(H_{0}: \mu=18\) (b) \(H_{0}: \mu=16.5\). (c) \(H_{0}: \mu<18\).
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