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Chapter 10: Q. 22 (page 669)

A random sample of size n will be selected from a population, and the proportion of those in the sample who have a Facebook page will be calculated. How would the margin of error for a 95%confidence interval be affected if the sample size were increased from 50to 200?

(a) It remains the same.

(b) It is multiplied by 2.

(c) It is multiplied by 4.

(d) It is divided by 2.

(e) It is divided by 4

Short Answer

Expert verified

It would be divided by 2.

Step by step solution

01

Step 1. Given information

It is given that the sample size is increased from 50to 200.

Confidence level is95%.

02

Step2. Simplify

The formula to compute the margin of error is:E=Zα2×p^(1-p^)n

Here,

n is the sample size.

p^is the sample proportion.

E is the margin error.

THE above mentioned formula indicates that there is an inverse relationship between the margin of error and the sample size.

If sample size increases from 50 to 200 then, it means that there is 4 times increase in the sample size. So, the formula can be written as:

E=Z0.052×p^(1-p^)4n=12×Z0.052×p^(1-p^)n

So, it could be concluded that the new margin of error is half of the old margin of error and it has been divided by 2.

Hence, the correct option is (d).

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Most popular questions from this chapter

A driving school wants to find out which of its two instructors is more effective at preparing students to pass the state’s driver’s license exam. An incoming class of 100students is randomly assigned to two groups, each of size 50. One group is taught by Instructor A; the other is taught by Instructor B. At the end of the course, 30of Instructor A’s students and 22of Instructor B’s students pass the state exam. Do these results give convincing evidence that Instructor A is more effective?

Min Jae carried out the significance test shown below to answer this question. Unfortunately, he made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one.

State: I want to perform a test of

H0:p1-p2=0

Ha:p1-p2>0

where p1=the proportion of Instructor A's students that passed the state exam and p2=the proportion of Instructor B's students that passed the state exam. Since no significance level was stated, I'll use σ=0.05

Plan: If conditions are met, I’ll do a two-sample ztest for comparing two proportions.

Random The data came from two random samples of 50students.

- Normal The counts of successes and failures in the two groups -30,20,22, and 28-are all at least 10.

- Independent There are at least 1000 students who take this driving school's class.

Do: From the data, p^1=2050=0.40and p^2=3050=0.60. So the pooled proportion of successes is

p^C=22+3050+50=0.52

- Test statistic

localid="1650450621864" z=(0.40-0.60)-00.52(0.48)100+0.52(0.48)100=-2.83

- p-value From Table A, localid="1650450641188" P(z≥-2.83)=1-0.0023=0.9977.

Conclude: The p-value, 0.9977, is greater than α=0.05, so we fail to reject the null hypothesis. There is no convincing evidence that Instructor A's pass rate is higher than Instructor B's.

A scatterplot and a least-squares regression line are shown in the figure below. If the labeled point P(20,24)is removed from the data set, which of the following statements is true ?

(a) The slope will decrease, and the correlation will decrease.

(b) The slope will decrease, and the correlation will increase.

(c) The slope will increase, and the correlation will increase.

(d) The slope will increase, and the correlation will decrease.

(e) No conclusion can be drawn since the other coordinates are unknown.

Do drivers reduce excessive speed when they encounter police radar? Researchers studied the behavior of a sample of drivers on a rural interstate highway in Maryland where the speed limit was 55miles per hour. They measured the speed with an electronic device hidden in the pavement and, to eliminate large trucks, considered only vehicles less than 20feet long. During some time periods (determined at random), police radar was set up at the measurement location. Here are some of the data.

Number of vehiclesNumber over65mphNo radar12,9315,690Radar3,2851,051

(a) The researchers chose a rural highway so that cars would be separated rather than in clusters because some cars might slow when they see other cars slowing. Explain why this is important.

(b) Does the proportion of speeding drivers differ significantly when the radar is being used and when it isn’t? Use information from the Minitab computer output below to support your answer.

Mrs. Woods and Mrs. Bryan are avid vegetable gardeners. They use different fertilizers, and each claims that hers is the best fertilizer to use when growing tomatoes. Both agree to do a study using the weight of their tomatoes as the response variable. They had each planted the same varieties of tomatoes on the same day and fertilized the plants on the same schedule throughout the growing season. At harvest time, they each randomly select 15tomatoes from their respective gardens and weigh them. After performing a two-sample t-test on the difference in mean weights of tomatoes, they get t=5.24and P=0.0008. Can the gardener with the larger mean claim that her fertilizer caused her tomatoes to be heavier?

(a) No, because the soil conditions in the two gardens is a potential confounding variable.

(b) No, because there was no replication.

(c) Yes, because a different fertilizer was used on each garden.

(d) Yes, because random samples were taken from each garden.

In an experiment to learn whether Substance M can help restore memory, the brains of 20rats were treated to damage their memories. The rats were trained to run a maze. After a day, 10rats (determined at random) were given M and 7of them succeeded in the maze. Only 2of the 10control rats were successful. The two-sample z test for “no difference” against “a significantly higher proportion of the M group succeeds”

(a) gives z=2.25,P<0.02

(b) gives z=2.60,P<0.005

(c) gives z=2.25,P<0.04but not <0.02

(d) should not be used because the Random condition is violated

(e) should not be used because the Normal condition is violated.
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