/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 The article "Religion and Well-B... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 87

The article "Religion and Well-Being Among Canadian University Students: The Role of Faith Groups on Campus" (Journal for the Scientific Study of Religion \([1994]: 62-73\) ) compared the self-esteem of students who belonged to Christian clubs and students who did not belong to such groups. Each student in a random sample of \(n=169\) members of Christian groups (the affiliated group) completed a questionnaire designed to measure self-esteem. The same questionnaire was also completed by each student in a random sample of \(n=124\) students who did not belong to a religious club (the unaffiliated group). The mean self-esteem score for the affiliated group was \(25.08\), and the mean for the unaffiliated group was \(24.55 .\) The sample standard deviations weren't given in the article, but suppose that they were 10 for the affiliated group and 8 for the unaffiliated group. Is there evidence that the true mean self- esteem score differs for affiliated and unaffiliated students? Test the relevant hypotheses using a significance level of \(.01\).

Short Answer

Expert verified
Based on the hypothesis testing with a significance level of 0.01, there was insufficient evidence to conclude that the true mean self-esteem score differs between the affiliated and unaffiliated groups. The test statistic (~0.553) did not exceed the critical values (~卤2.58), hence the null hypothesis was not rejected.

Step by step solution

01

Set up the hypotheses

The null hypothesis (H0) is that the true mean self-esteem scores of the two groups are equal, and the alternative hypothesis (Ha) is that they are not equal. That is: \(H0: 碌1 = 碌2\) and \(Ha: 碌1 鈮� 碌2\) where 碌1 and 碌2 represent the true mean self-esteem scores of the affiliated and unaffiliated group, respectively.
02

Calculate the standard error

The standard error of the difference between the two sample means can be calculated using the formula \(\sqrt{{s1^2/n1 + s2^2/n2}}\), where \(s1 = 10\) (the standard deviation of the affiliated group), \(s2 = 8\) (the standard deviation of the unaffiliated group), \(n1 = 169\) (the sample size of the affiliated group), and \(n2 = 124\) (the sample size of the unaffiliated group). This equals to approximately 0.958.
03

Calculate the test statistic

The test statistic is calculated as the observed difference in sample means, divided by the standard error. Here, the observed difference in sample means (\(x1 - x2\)) is \(25.08 - 24.55 = 0.53\). So the test statistic is \(0.53/0.958 \approx 0.553\).
04

Determine the critical value

Using a significance level of 0.01, and given that this is a two-tailed test (because the alternative hypothesis is that the means are 'not equal'), the critical value from the t-distribution table is approximately 卤2.58 (using degrees of freedom \(df = min(n1-1,n2-1) = min(168,123) = 123\)).
05

Make a decision

Since the test statistic does not fall in the critical region (i.e., it is not less than -2.58 or greater than 2.58), we will fail to reject the null hypothesis, meaning there is insufficient evidence at this significance level to conclude that the true mean self-esteem score differs for affiliated and unaffiliated students.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Self-Esteem Measurement
Self-esteem measurement is an essential tool in psychological research, which gauges an individual's overall perception of their value or worth. In studies like the one mentioned, self-esteem is typically assessed using validated questionnaires or scales. These tools involve statements related to personal traits, feelings, or behaviors, to which participants respond in a structured manner, often using a Likert scale.

The results, which are numerical scores, help quantify the level of self-esteem among individuals or groups. When comparing different groups, as in our exercise, these scores can be used to find differences in self-esteem levels among members of Christian clubs versus those who are unaffiliated with religious groups. It's important to remember that the validity of the conclusions drawn from self-esteem measurements depends on the questionnaire's reliability and the representativeness of the sample used.
Significance Level
The significance level is a crucial concept in hypothesis testing that defines the threshold for determining when to reject the null hypothesis. In this exercise, the significance level is set at 0.01. This means there is a 1% risk of rejecting the null hypothesis when it is actually true, also known as a Type I error.

Choosing a 0.01 significance level indicates a more stringent standard for evidence compared to the commonly used 0.05 level. Researchers set this threshold before conducting the test to avoid bias in interpretation. In simpler terms, it helps in deciding how much "proof" is needed to say that a difference between groups is statistically significant, rather than due to random chance.
t-Distribution
The t-distribution is a probability distribution often used in hypothesis testing, particularly when dealing with small sample sizes or unknown population variances. It resembles the normal distribution but has thicker tails, which makes it more accommodating of outliers or sample variance.

For our exercise, the t-distribution was used to obtain a critical value, which in this case is 卤2.58 for a significance level of 0.01. This value serves as a cutoff in a two-tailed test to determine whether the test statistic calculated from the sample falls into the rejection region, calling into question the null hypothesis. As sample size increases, the t-distribution closely resembles the standard normal distribution, which is used when we have exact population parameters.
Two-Sample t-Test
A two-sample t-test is a statistical method used to compare the means of two independent groups and determine if there is a statistically significant difference between them. In the example provided, researchers wanted to know if the self-esteem scores of students in Christian clubs were different from those of unaffiliated students.

This test involves several steps:
  • Formulating null and alternative hypotheses.
  • Calculating the standard error of the mean difference.
  • Computing the test statistic.
  • Comparing this statistic to the critical value from the t-distribution.

In this scenario, when the test statistic (0.553) does not exceed the critical value (卤2.58), we fail to reject the null hypothesis. This means there isn't enough statistical evidence at a 0.01 significance level to say that the self-esteem scores differ between the two groups.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The article "The Sorority Rush Process: SelfSelection, Acceptance Criteria, and the Effect of Rejection" (Journal of College Student Development [1994]: \(346-353\) ) reported on a study of factors associated with the decision to rush a sorority. Fifty-four women who rushed a sorority and 51 women who did not were asked how often they drank alcoholic beverages. For the sorority rush group, the mean was \(2.72\) drinks per week and the standard deviation .86. For the group who did not rush, the mean was \(2.11\) and the standard deviation \(1.02 .\) Is there evidence to support the claim that those who rush a sorority drink more than those who do not rush? Test the relevant hypotheses using \(\alpha=.01\). What assumptions are required in order for the two-sample \(t\) test to be appropriate?

The article "More Students Taking AP Tests" (San Luis Obispo Tribune, January 10,2003 ) provided the following information on the percentage of students in grades 11 and 12 taking one or more AP exams and the percentage of exams that earned credit in 1997 and 2002 for seven high schools on the central coast of California. a. Assuming it is reasonable to regard these seven schools as a random sample of high schools located on the central coast of California, carry out an appropriate test to determine if there is convincing evidence that the mean percentage of exams earning college credit at central coast high schools declined between 1997 and \(2002 .\) b. Do you think it is reasonable to generalize the conclusion of the test in Part (a) to all California high schools? Explain. c. Would it be reasonable to use the paired \(t\) test with the data on percentage of students taking one or more AP classes? Explain.

British health officials have expressed concern about problems associated with vitamin D deficiency among certain immigrants. Doctors have conjectured that such a deficiency could be related to the amount of fiber in a person's diet. An experiment was designed to compare the vitamin D plasma half-life for two groups of healthy individuals. One group was placed on a normal diet, whereas the second group was placed on a high-fiber diet. The accompanying table gives the resulting data (from "Reduced Plasma Half-Lives of Radio-Labeled \(25(\mathrm{OH}) \mathrm{D} 3\) in Subjects Receiving a High-Fibre Diet," British Journal of Nutrition [1993]: 213-216). \(\begin{array}{lllllll}\text { Normal diet } & 19.1 & 24.0 & 28.6 & 29.7 & 30.0 & 34.8\end{array}\) \(\begin{array}{llllllll}\text { High-fiber diet } & 12.0 & 13.0 & 13.6 & 20.5 & 22.7 & 23.7 & 24.8\end{array}\) Use the following MINITAB output to determine whether the data indicate that the mean half-life is higher for those on a normal diet than those on a high- fiber diet. Assume that treatments were assigned at random and the two plasma half-life distributions are normal. Test the appropriate hypotheses using \(\alpha=.01 .\) Two-sample \(\mathrm{T}\) for normal vs high Iw \(\begin{array}{lccrr} & \mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ \text { Normal } & 6 & 27.70 & 5.44 & 2.2 \\ \text { High } & 7 & 18.61 & 5.55 & 2.1\end{array}\) \(5.2\) 15 High \(95 \%\) C.l. for mu normal - mu high: \((2.3,15.9)\) T-Test mu normal = mu high(vs >):T \(=2.97 \mathrm{P}=0.0070 \mathrm{DF}=10\)

Do girls think they don't need to take as many science classes as boys? The article "Intentions of Young Students to Enroll in Science Courses in the Future: An Examination of Gender Differences" (Science Education [1999]: \(55-76\) ) gives information from a survey of children in grades 4,5, and \(6 .\) The 224 girls participating in the survey each indicated the number of science courses they intended to take in the future, and they also indicated the number of science courses they thought boys their age should take in the future. For each girl, the authors calculated the difference between the number of science classes she intends to take and the number she thinks boys should take. a. Explain why these data are paired. b. The mean of the differences was \(-.83\) (indicating girls intended, on average, to take fewer classes than they thought boys should take), and the standard deviation was 1.51. Construct and interpret a \(95 \%\) confidence interval for the mean difference.

An individual can take either a scenic route to work or a nonscenic route. She decides that use of the nonscenic route can be justified only if it reduces true average travel time by more than \(10 \mathrm{~min}\). a. If \(\mu_{1}\) refers to the scenic route and \(\mu_{2}\) to the nonscenic route, what hypotheses should be tested? b. If \(\mu_{1}\) refers to the nonscenic route and \(\mu_{2}\) to the scenic route, what hypotheses should be tested?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.