Statistics for Business and Economics (13th Edition)
Statistics for Business and Economics (13th Edition)
13th Edition
ISBN: 9780134506593
Author: James T. McClave, P. George Benson, Terry Sincich
Publisher: PEARSON
bartleby

Concept explainers

bartleby

Videos

Textbook Question
Book Icon
Chapter 5, Problem 5.52LM

The standard deviation (or, as it is usually called , the standard error) of the sampling distribution for the sample mean, x ¯ , is equal to the standard deviation of the population from which the sample was selected, divided by the square root of the sample size. That is,

σ A = σ n 3

  1. a. As the sample size is increased, what happens to the standard error of x ¯ ? Why is this property considered important?
  2. b. Suppose a sample statistic has a standard error that is not a function of the sample size. In other words, the standard error remains constant as n changes. What would this imply about the statistic as an estimator of a population parameter?
  3. c. Suppose another unbiased estimator (call it A) of the population mean is a sample statistic with a standard error equal to

    σ A = σ n 3

    Which of the sample statistics, x ¯ or A, is preferable as an estimator of the population mean? Why?

  4. d. Suppose that the population standard deviation σ is equal to 10 and that the sample size is 64. Calculate the standard errors of x ¯ and A. Assuming that the sampling distribution of A is approximately normal, interpret the standard errors. Why is the assumption of (approximate) normality unnecessary for the sampling distribution of x ¯ ?

a.

Expert Solution
Check Mark
To determine

To describe: what happens to the standard error of x¯ when sample size increases.

To explain: Why is this property considered important.

Answer to Problem 5.52LM

The standard deviation (standard error) decreases when sample size increases.

The property is important because the estimate variable would be less when the sample size is large.

Explanation of Solution

Calculation:

Sampling distribution of x¯ :

When a random sample of size n is selected from a population which is normally distributed, then the sampling distribution of x¯ is also normally distributed.

Properties:

  • The mean of sampling distribution is equal to the mean of the population.

    μx¯=E(x)=μ

  • The standard deviation of the sampling distribution is equal to the ratio of the population standard deviation and square root of sample size.

    σx¯=σn

  • When the sample size n is or more of the population size N then the σn must be multiplied by the finite correction factor (Nn)(N1) .
  • For most sampling situations, the correction factor is close to 1 and is ignored.
  • The standard deviation σx¯ sometimes referred as the standard error of the mean.

As the sample size increases the standard deviation (standard error) decreases because the sample size and the standard deviation of x¯ are inversely proportional.

This property is important because the estimate variable would be less when the sample size is large.

b.

Expert Solution
Check Mark
To determine

To explain: About the statistic as an estimator of a population parameter when the standard error remains constant as n changes.

Explanation of Solution

If the standard error remains constant as n changes then it would not be a good estimator for the parameter. When the standard error is not a function of the sample size then the statistic based on one observation is as good estimator as the statistic based on 1,000 observations.

c.

Expert Solution
Check Mark
To determine

To explain: Which of the sample statistic x¯or A is preferable as an estimator of the population.

Explanation of Solution

Given info:

An unbiased estimator A of a population mean is a sample statistic with a standard error equal to σA=σn3 is considered.

Justification:

The statistic x¯ should be preferred than A as an estimator for the population mean because the standard error of x¯ is fewer than the standard error of A.

d.

Expert Solution
Check Mark
To determine

To find: The standard errors of x¯ and A.

To explain: Whether the assumption of normality is unnecessary for the sampling distribution of x¯ .

Answer to Problem 5.52LM

The standard error of x¯ is 1.25.

The standard error of A is 2.5.

Explanation of Solution

Given info:

The population standard deviation σ is 10 and the sample size is 64. Consider the sampling distribution of A is approximately normal.

Calculation:

The standard error of x¯ is,

σx¯=σn

Substitute σ as 10 and n as 64 in the formula,

σx¯=σn=1064=108=1.25

Thus, the standard error of x¯ is 1.25.

Central Limit Theorem:

Central limit theorem states that the sampling distribution of x¯ with mean μx¯=μ and standard deviation σx¯=σn would be approximately normal as the sample size is sufficiently large.

The normal approximation to the sampling distribution of x¯ would be better when the sample size is larger, n=64 .

Therefore the sampling distribution of x¯ is approximately normal with mean μx¯=μ and standard deviation σx¯=1.25 .

Empirical rule:

It the distributions of a data set are approximately symmetric or bell shaped then the standard deviations have the following features,

  • About 68% of the data falls within one standard deviation of the mean.
  • About 95% of the data falls within two standard deviations of the mean.
  • About 99.7% of the data falls within three standard deviations of the mean.

One standard deviation from mean is,

μx¯±σx¯=μ±1.25=(μ1.25,μ+1.25)

Two standard deviations from mean is,

μx¯±2σx¯=μ±2(1.25)=μ±2.50=(μ2.50,μ+2.50)

Three standard deviations from mean is,

μx¯±3σx¯=μ±3(1.25)=μ±3.75=(μ3.75,μ+3.75)

From the empirical rule 68% of the data falls within (μ1.25,μ+1.25) , 95% of the data falls within (μ2.50,μ+2.50) , and 99.7% of the data falls within (μ3.75,μ+3.75) .

The standard error of A is,

σA=σn3

Substitute σ as 10 and n as 64 in the formula,

σA=σn3=10643=104=2.5

Thus, the standard error of A is 2.5.

One standard deviation from mean is,

μA±σA=μ±2.50=(μ2.50,μ+2.50)

Two standard deviations from mean is,

μA±2σA=μ±2(2.50)=μ±5.00=(μ5.00,μ+5.00)

Three standard deviations from mean is,

μA±3σA=μ±3(2.50)=μ±7.50=(μ7.50,μ+7.50)

From the empirical rule 68% of the data falls within (μ2.50,μ+2.50) , 95% of the data falls within (μ5.00,μ+5.00) , and 99.7% of the data falls within (μ7.50,μ+7.50) .

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Chapter 5 Solutions

Statistics for Business and Economics (13th Edition)

Ch. 5.2 - Refer to Exercise 5.5, in which we found the...Ch. 5.3 - Will the sampling distribution of x always be...Ch. 5.3 - Suppose a random sample of n = 25 measurements is...Ch. 5.3 - Suppose a random sample of n measurements is...Ch. 5.3 - A random sample of n = 64 observations is drawn...Ch. 5.3 - Refer to Exercise 5.18. Find the probability that...Ch. 5.3 - A random sample of n = 900 observations is...Ch. 5.3 - A random sample of n = 100 observations is...Ch. 5.3 - Open the applet Sampling Distributions. On the...Ch. 5.3 - Open the applet Sampling Distributions. On the...Ch. 5.3 - Voltage sags and swells. Refer to the Electrical...Ch. 5.3 - Salary of a travel management professional....Ch. 5.3 - Corporate sustainability of CPA firms. Refer to...Ch. 5.3 - Critical-part failures in NASCAR vehicles. Refer...Ch. 5.3 - Tomato as a taste modifier. Miraculin is a protein...Ch. 5.3 - Prob. 5.28ACICh. 5.3 - Levelness of concrete slabs. Geotechnical...Ch. 5.3 - Video game players and divided attention tasks....Ch. 5.3 - Exposure to a chemical in Teflon-coated cookware....Ch. 5.3 - Rental car fleet evaluation. National Car Rental...Ch. 5.3 - Prob. 5.34ACACh. 5.3 - Handwashing vs. handrubbing. The British Medical...Ch. 5.4 - Suppose a random sample of n measurements is...Ch. 5.4 - Suppose a random sample of n = 500 measurements is...Ch. 5.4 - A random sample of n = 80 measurements is drawn...Ch. 5.4 - A random sample of n = 250 measurements is drawn...Ch. 5.4 - A random sample of n = 1, 500 measurements is...Ch. 5.4 - Consider a population with values of x equal to 0...Ch. 5.4 - Dentists use of laughing gas. According to the...Ch. 5.4 - Cable TV subscriptions and cord cutters ....Ch. 5.4 - Do social robots walk or roll? Refer to the...Ch. 5.4 - Working on summer vacation. According to a Harris...Ch. 5.4 - Hospital work-related injuries. According to an...Ch. 5.4 - Hotel guest satisfaction. Refer to the results of...Ch. 5.4 - Stock market participation and IQ. Refer to The...Ch. 5.4 - Fingerprint expertise. Refer to the Psychological...Ch. 5.4 - Who prepares your tax return? As part of a study...Ch. 5.4 - Apps not working on smartphone. In a survey titled...Ch. 5 - The standard deviation (or, as it is usually...Ch. 5 - Consider a sample statistic A. As with all sample...Ch. 5 - A random sample of 40 observations is to be drawn...Ch. 5 - A random sample of n = 68 observations is selected...Ch. 5 - A random sample of n = 500 observations is...Ch. 5 - A random sample of n = 300 observations is...Ch. 5 - Use a statistical software package to generate 100...Ch. 5 - Use a statistical software package to generate 100...Ch. 5 - Suppose x equals the number of heads observed when...Ch. 5 - A random sample of size n is to be drawn from a...Ch. 5 - Requests to a Web server. In Exercise 4.175 (p....Ch. 5 - Improving SAT scores. Refer to the Chance (Winter...Ch. 5 - Study of why EMS workers leave the job. A study of...Ch. 5 - Downloading apps to your cell phone. Refer to...Ch. 5 - Surface roughness of pipe. Refer to the...Ch. 5 - Analysis of supplier lead time. Lead time is the...Ch. 5 - Producing machine bearings. To determine whether a...Ch. 5 - Quality control. Refer to Exercise 5.68. The mean...Ch. 5 - Length of job tenure. Researchers at the Terry...Ch. 5 - Switching banks after a merger. Banks that merge...Ch. 5 - Piercing rating of fencing safety jackets. A...Ch. 5 - Errors in filling prescriptions A large number of...Ch. 5 - Purchasing decision. A building contractor has...Ch. 5 - Motivation of drug dealers. Refer to the Applied...Ch. 5 - Soft-drink bottles. A soft-drink bottler purchases...
Knowledge Booster
Background pattern image
Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Statistics 4.1 Point Estimators; Author: Dr. Jack L. Jackson II;https://www.youtube.com/watch?v=2MrI0J8XCEE;License: Standard YouTube License, CC-BY
Statistics 101: Point Estimators; Author: Brandon Foltz;https://www.youtube.com/watch?v=4v41z3HwLaM;License: Standard YouTube License, CC-BY
Central limit theorem; Author: 365 Data Science;https://www.youtube.com/watch?v=b5xQmk9veZ4;License: Standard YouTube License, CC-BY
Point Estimate Definition & Example; Author: Prof. Essa;https://www.youtube.com/watch?v=OTVwtvQmSn0;License: Standard Youtube License
Point Estimation; Author: Vamsidhar Ambatipudi;https://www.youtube.com/watch?v=flqhlM2bZWc;License: Standard Youtube License