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Chapter 7: Q. 64 (page 480)

The distribution of grade point average for students at a large high school is skewed to the left with a mean of 3.53 and a standard deviation of 1.02.

a. Describe the shape of the sampling distribution of x炉 for SRSs of size n = 4 from the population of students at this high school. Justify your answer.

b. Describe the shape of the sampling distribution of x炉 for SRSs of size n = 50 from the Page Number: 480 population of students at this high school. Justify your answer.

Short Answer

Expert verified

a. The sample mean's sampling distribution is skewed to the left.

b. The sampling distribution of the sample mean is approximately normal.

Step by step solution

01

Part (a) : Step 1 : Given information 

Given :

Mean :3.53

Standard deviation :1.02.

Size n=4

02

Part (a) : Step 2 : Simplification 

The distribution of the population is skewed to the left.

n=4

Theorem of the center limit: The sampling distribution of the sample mean x is roughly normal if the sample size is 30or greater. The central limit theorem cannot be applied since the sample size of four is less than thirty. The sample mean's sampling distribution has the same shape as the population distribution in this example, hence the sample mean's sampling distribution is skewed to the left.

03

Part (b) : Step 1 : Given information 

Given :

Mean :3.53

Standard deviation : 1.02.

Sizen=50

04

Part (b) : Step 2 : Simplification 

The distribution of the population is skewed to the left.

n=50people

Theorem of the center limit: The sampling distribution of the sample mean is roughly normal if the sample size is 30or greater. Because the sample size of 50is greater than 30, the central limit theorem can be used, and the sampling distribution of the sample mean is approximately normal.

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Most popular questions from this chapter

Even more tall girls, Refer to Exercises 12and 14. Suppose that the sample mean height of the twenty 16-year-old females is x=65.8inches. Would this sample mean provide convincing evidence that the average height of all 16-year-old females at this school is greater than 64inches? Explain your reasoning.

You work for an advertising agency that is preparing a new television commercial to appeal to women. You have been asked to design an experiment to compare the effectiveness of three versions of the commercial. Each subject will be shown one of the three versions and then asked to reveal her attitude toward the product. You think there may be large differences in the responses of women who are employed and those who are not. Because of these differences, you should use

a. a block design, but not a matched pairs design.

b. a completely randomized design.

c. a matched pairs design.

d. a simple random sample.

e. a stratified random sample.

More homework Some skeptical Ap庐 Statistics students want to investigate the newspaper's claim in Exercise 11, so they choose an SRS of 100students from the school to interview. In their sample, 45students completed their homework last week. Does this provide convincing evidence that less than 60%of all students at the school completed their assigned homework last week?

a. What is the evidence that less than 60%of all students completed their assigned homework last week?

b. Provide two explanations for the evidence described in part (a).

We used technology to simulate choosing 250SRSs of size n=100n=100from a population of 2000students where 60%completed their assigned homework last week. The dotplot shows pp^the sample proportion of students who completed their assigned homework last week for each of the 250simulated samples.

c. There is one dot on the graph at 0.73. Explain what this value represents.

d. Would it be surprising to get a sample proportion of p=0.45p^=0.45or smaller in an SRS of size 100when p=0.60p=0.60? Justify your answer.

e. Based on your previous answers, is there convincing evidence that less than 60%of all students at the school completed their assigned homework last week? Explain your reasoning.

Sample minimums List all 6possible SRSS of size n=2, calculate the minimum age for each sample, and display the sampling distribution of the sample minimum on a dotplot. Is the sample minimum an unbiased estimator of the population minimum? Explain your answer.

COLORAGE
RED1
WHITE5
SILVER8
RED20

A study of rush-hour traffic in San Francisco counts the number of people in each car entering a freeway at a suburban interchange. Suppose that this count has mean 1.6 and standard deviation 0.75 in the population of all cars that enter at this interchange during rush hour.

a. Without doing any calculations, explain which event is more likely:

  • randomly selecting 1 car entering this interchange during rush hour and finding 2 or more people in the car
  • randomly selecting 35 cars entering this interchange during rush hour and finding an average of 2 or more people in the cars

b. Explain why you cannot use a Normal distribution to calculate the probability of the first event in part (a).

c. Calculate the probability of the second event in part (a).
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