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Chapter 7: Q. 70 (page 481)

What does the CLT say? Asked what the central limit theorem says, a student replies, "As you take larger and larger samples from a population, the variability of the sampling distribution of the sample mean decreases." Is the student right? Explain your answer.

Short Answer

Expert verified

The statement made by the student is Yes, correct

Step by step solution

01

Given information 

The student made a statement "The variability of the sampling distribution of the sample mean diminishes as you take more and larger samples from a population."

02

Explanation 

If the sample size is higher than or equal to 30, the sampling distribution of the sample mean will be nearly normal with mean and standard deviation, according to the central limit theorem is 蟽/n.

The standard deviation of the sampling distribution drops as the value of n increases, and the spread of the sampling distribution of the sample mean decreases.

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Most popular questions from this chapter

According to the U.S. Census, the proportion of adults in a certain county who owned their own home was 0.71. An SRS of 100 adults in a certain section of the county found that 65 owned their home. Which one of the following represents the approximate probability of obtaining a sample of 100 adults in which 65 or fewer own their home, assuming that this section of the county has the same overall proportion of adults who own their home as does the entire county?

a. (10065)(0.71)65(0.29)3510065(0.71)65(0.29)35

b. (10065)(0.29)65(0.71)3510065(0.29)65(0.71)35

c.

P(z鈮�0.65-0.71(0.71)(0.29)100)Pz鈮�0.65-0.71(0.71)(0.29)100

d.P(z鈮�0.65-0.71(0.65)(0.35)100)Pz鈮�0.65-0.71(0.65)(0.35)100

e.P(z鈮�0.65-0.71(0.71)(0.29)100)Pz鈮�0.65-0.71(0.71)(0.29)100

Squirrels and their food supply (3.2) Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over 16years.4Computer output from a least-squares 4regression on these data and a residual plot are shown here

a. Is a linear model appropriate for these data? Explain.

b. Give the equation for the least-squares regression line. Define any variables you use.

c. Interpret the values of r2and sin context.

AP2.20 A grocery chain runs a prize game by giving each customer a ticket that may win a prize when the box is scratched off. Printed on the ticket is a dollar value ( \( 500, \) 100, \(25) or the statement "This ticket is not a winner." Monetary prizes can be redeemed for groceries at the store. Here is the probability distribution of the amount won on a randomly selected ticket:

Which of the following are the mean and standard deviation, respectively, of the winnings?

a. \) 15.00, \( 2900.00

b.\) 15.00, \( 53.85

c. \) 15.00, \( 26.93

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More sample proportions List all 4possible SRSs of size n=3, calculate the proportion of red cars in the sample, and display the sampling distribution of the sample proportion on a dot plot with the same scale as the dot plot in Exercise 19. How does the variability of this sampling distribution compare with the variability of the sampling distribution from Exercise 19? What does this indicate about increasing the sample size?

From exercise19:

Car NumberColorAge
1
Red
1
2
White
5
3
Silver
8
4
Red
20

Increasing the sample size of an opinion poll will reduce the

a. bias of the estimates made from the data collected in the poll.

b. variability of the estimates made from the data collected in the poll.

c. effect of nonresponse on the poll.

d. variability of opinions in the sample.

e. variability of opinions in the population.
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