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Chapter 8: Q.3.2 (page 494)

1. In the company鈥檚 prior-year survey, 80% of customers surveyed said they were 鈥渟atisfied鈥� or 鈥渧ery satisfied.鈥� Using this value as a guess for p藛, find the sample size needed for a margin of error of 3% at a 95% confidence level.

What if the company president demands 99% confidence instead? Determine how this would affect your answer to Question 1.

Short Answer

Expert verified

The required sample size is 1180.

From the calculations, we know that the increase in confidence level is leading to an increase in the sample size.

Step by step solution

01

Given Information

Given that

Population proportion(p^)=80%=0.80

Margin of error(E)=3%=0.03

Confidence level=95%

02

Explanation

From the standard normal table, thez-score at 99%the confidence level is 2.579

The sample size is calculated as:

n=(p^)(1-p^)zE2

=0.80(1-0.80)2.5760.032

=1179.537

鈮�1180

Thus, the required sample size is 1180.

From the calculations, we know that the increase in confidence level is leading to an increase in the sample size.

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Most popular questions from this chapter

NAEP scores Refer to Exercise 5. Below your sketch, choose one value of x inside the shaded region and draw its corresponding confidence interval. Do the same for one value of x outside the shaded region. What is the most important difference between these intervals?

You have measured the systolic blood pressure of an SRS of 25company employees. A 95%confidence interval for the mean systolic blood pressure for the employees of this company is (122,138). Which of the following statements gives a valid interpretation of this interval?

(a) 95%of the sample employees have systolic blood pressure between 122and138.

(b) 95%of the population of employees have systolic blood pressure between 122and138.

(c) If the procedure were repeated many times, 95%of the resulting confidence intervals would contain the population mean systolic blood pressure.

(d) The probability that the population mean blood pressure is between 122and138is 0.95.

(e) If the procedure were repeated many times, 95%of the sample means would be between122and138.

Scientists collect data on the blood cholesterol levels (milligrams per deciliter of blood) of a random sample of 24laboratory rats. A 95%confidence interval for the mean blood cholesterol level Mis 80.2to 89.8. Which of the following would cause the most worry about the validity of this interval?

(a) There is a clear outlier in the data.

(b) A stemplot of the data shows a mild right skew.

(c) You do not know the population standard deviation S.

(d) The population distribution is not exactly Normal.

(e) None of these would be a problem because the t procedures are robust.

The Gallup Poll interviews 1600people. Of these, 18%say that they jog regularly. The news report adds: "The poll had a margin of error of plus or minus three percentage points at a 95%confidence level." You can safely conclude that

(a) 95%of all Gallup Poll samples like this one give answers within卤3%of the true population value.

(b) the percent of the population who jog is certain to be between 15%and21%.

(c) 95%of the population jog between5%and21%of the time.

(d) we can be 95%confident that the sample proportion is captured by the confidence interval.

(e) if Gallup took many samples,95%of them would find that 18%of the people in the sample jog.

5. NAEP scores Young people have a better chance of full-time employment and good wages if they are good with numbers. How strong are the quantitative skills of young Americans of working age? One source
of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a
short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0to 500. For example, a person who scores 233can add the amounts of two checks appearing on a bank deposit slip; someone scoring 325can determine the price of a meal from a menu; a person scoring 375 can
transform a price in cents per ounce into dollars per pound. Suppose that you give the NAEP test to an SRS of 840people from a large population in which the scores have mean 280and standard deviation蟽=60. The mean xof the 840 scores will vary if you take repeated samples.
(a) Describe the shape, center, and spread of the sampling distribution of x.
(b) Sketch the sampling distribution of x. Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the68鈥�95鈥�99.7rule, about 95%of all values of x炉lie within a distance mof the mean of the sampling distribution. What is m? Shade the region on the axis of your sketch that is within m of the mean.

(d) Whenever x炉falls in the region you shaded, the population mean 渭lies in the confidence intervalx卤m. For what percent of all possible samples does the interval capture 渭?

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