/*! 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} Q 7.64. A variable of a population has m... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q 7.64. (page 307)

A variable of a population has mean 渭 and standard deviation蟽. that For a large sample size n, answer the following questions.

a. Identify the distribution ofx.

b. Does your answer to part (a) depend on n being large? Explain your answer.

c. Identify the mean and the standard deviation ofx.

d. Does your answer to part (c) depend on the sample size being large? Why or why not?

Short Answer

Expert verified

Part a) By application of CLT, sample mean follows Normal distribution.

Part b) Yes, if the sample is not a large sample then we cannot apply CLT to approximate the distribution of the sample mean as Normal distribution.

Part c) Mean of x炉=渭And standard deviation of x炉=蟽n.

Part d) No, the expression for mean and S. d of X炉remains same the large samples also.

Step by step solution

01

Given information

Population variable has the mean 渭and standard deviation蟽

Sample size nis large.

02

Part a) Step 1:

By application of CLT, sample mean follows Normal distribution.

03

Part b) Step 1:

Yes, since we can approximate the distribution of sample mean by Normal distribution in case of unknown or Non normal population distribution iff the sample size nis large. That is if the sample is not a large sample then we cannot apply CLT to approximate the distribution of the sample mean as Normal distribution.

04

Part c) Step 1:

Mean of x炉=渭and standard deviation ofx炉=蟽n.

05

Part d) Step 1:

No, the expression for mean and S. d of X炉remains same the large samples also.

i.e. For large sample also渭x炉=渭and蟽x炉=蟽n.

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!

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

In Exercises 7.3-7.10, we have given population data for a variable. For each exercise, do the following tasks.
a. Find the mean, 渭, of the variable.
b. For each of the possible sample sizes, construct a table similar to Table 7.2 on page 293 and draw a dotplot for the sampling distribution of the sample mean similar to Fig. 7.1 on page 293.
c. Construct a graph similar to Fig. 7.3 and interpret your results.
d. For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.
e. For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be 0.5or less (in magnitude), that is, that the absolute value of the difference between the sample mean and the population mean is at most 0.5.
7.4 Population data: 2,5,8.

Refer to Fig. 7.6on page 306 .

a. Why are the four graphs in Fig. 7.6(a) all centered at the same place?

b. Why does the spread of the graphs diminish with increasing sample size? How does this result affect the sampling error when you estimate a population mean, 渭by a sample mean, x~ ?

c. Why are the graphs in Fig. 7.6(a) bell shaped?

d. Why do the graphs in Figs. 7.6(b)and (c) become bell shaped as the sample size increases?

Although, in general, you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it?

7.67 Brain Weights. In 1905, R. Pearl published the article "Biometrical Studies on Man. 1. Variation and Correlation in Brain Weight" (Biometrika, Vol. 4, pp. 13-104). According to the study, brain weights of S wedish men are normally distributed with a mean of 1.40kg and a standard deviation of 0.11kg

a. Determine the sampling distribution of the sample mean for samples of size 3 Interpret your answer in terms of the distribution of all possible sample mean brain weights for samples of three Swedish men.

b. Repeat part (a) for samples of size 12

c. Construct graphs similar to those shown in Fig. 7.4on page 304 .

d. Determine the percentage of all samples of three Swedish men that have mean brain weights within 0.1kg of the population mean brain weight of 1.40kg. Interpret your answer in terms of sampling error.

e. Repeat part (d) for samples of size 12

Paint Durability. A paint manufacturer in Pittsburgh claims that his paint will last an average of 5 years. Assuming that paint life is normally distributed and has a standard deviation of 0.5 year. answer the following questions:

a. Suppose that you paint one house with the paint and that the paint lasts 4.5 years. Would you consider that evidence against the manufacturer's claim? (Hint: Assuming that the manufacturer's claim is correct, determine the probability that the paint life for a randomly selected house painted with the paint is 4.5 years or less.)

b. Suppose that you paint 10 houses with the paint and that the paint lasts an average of 4.5 years for the 10 houses. Would you consider that evidence against the manufacturer's claim?

c. Repeat part (b) if the paint lasts an average of 4.9 years for the 10 houses painted.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

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.