Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Q.10.61 (page 417)

The formula for the pooled variance, sp2, is given on page 407 Show that, if the sample sizes, n1 and n2, are equal, then sp2 is th mean of s12 and s22.

Short Answer

Expert verified

If the sample sizes, n1 and n2, are equal, then sp2 is the mean of s12 and s22.

Step by step solution

01

Given Information

The sample sizes of the two populations, n1 and n2, are the same.

02

Explanation

A test statistic is used when conducting a hypothesis test based on independent samples to compare the means of two populations with equal and unknown standard deviations. Individual sample variances of two populations, s12and s22, are gathered and pooled by weighting them according to their sample size or degree of freedom, n1and n2.

sp2=n1-1s12+n2-1s22n1+n2-2

03

Explanation

sp2incorporates information about both samples' variability into a single estimate of the population variance's common value. As a result, spis known as a pooled standard variance.

When n1=n2,

⇒sp2=n1-1s12+n1-1s22n1+n1-2

⇒sp2=n1-1s12+s222n1-2

⇒sp2=s12+s222

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Stressed-Out Bus Drivers. An intervention program designed by the Stockholm Transit District was implemented to improve the work conditions of the city's bus drivers. Improvements were evaluated by G. Evans et al., who collected physiological and psychological data for bus drivers who drove on the improved routes (intervention) and for drivers who were assigned the normal routes (control). Their findings were published in the article "Hassles on the Job: A Study of a Job Intervention with Urban Bus Drivers" (Journal of Organizational Behavior, Vol. 20, pp. 199-208). Following are data, based on the results of the study, for the heart rates, in beats per minute, of the intervention and control drivers.

a. At the 5%significance level, do the data provide sufficient evidence to conclude that the intervention program reduces mean heart rate of urban bus drivers in Stockholm? (Note; x1=67.90, s1=5.49,x¯2=66.81and s2=9.04.

b. Can you provide an explanation for the somewhat surprising results of the study?

c. Is the study a designed experiment or an observational study? plain your answer.

In the article "Sleep Apnea in Adults With Traumatic Brain Injury: A Preliminary Investigation" (Archives of Physical Medicine and Rehabilitation, Vol. 82, Issue 3, pp. 316321), J. Webster et al, investigated sleep-related breathing disorders in adults with traumatic brain injuries (TBI). The respiratory disturbance index (RDI), which is the number of apneic and hypopneic episodes per hour of sleep, was used as a measure of severity of sleep apnea. An RDI of 5 or more indicates sleep-related breathing disturbances. The RDIs for the females and males in the study are as follows.

Use the technology of your choice to answer the following questions. Explain your answers.

a. If you had to choose between the use of pooledt-procedures and nonpooled t-procedures here, which would you choose?

b. Is it reasonable to use the type of procedure that you selected in part (a)?

Phyllodes Tumors. Phyllodes tumors of the breast are rare tumors that represent less than one percent of growths in the breast. Researchers I. Youn et al. presented characteristics of phyllodes tumors in the article "Phyllodes Tumors of the Breast: Ultrasonographic Findings and Diagnostic Performance of Ultrasound-Guided Core Needle Biopsy" . The following table provides summary statistics for the sizes, in millimeters (mm), of independent samples of benign and malignant phyllodes tumors.

At the 1%significance level, do the data provide sufficient evidence to conclude that, on average, malignant phyllodes tumors are larger than benign phyllodes tumors?

Suppose that you want to perform a hypothesis test to compare the means of two populations, using independent simple random samples. Assume that the two distributions of the variable under consideration have the same shape, but are not normal, and both sample sizes are large. Answer the following questions and explain your answers.

a. Is it permissible to use the pooled t-test to perform the hypothesis test?

b. Is it permissible to use the Mann-Whitney test to perform the hypothesis test?

c. Which procedure is preferable, the pooled t-test or the Mann-Whitney test?

In each of exercise 10.13-10.18, we have presented a confidence interval for the difference,μ1andμ2, between two population means. interpret each confidence interval.

95%CI is from-20to-15
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.

Sign-up for free