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Chapter 6: Q44E (page 356)

A random sample of 50 consumers taste-tested a new snack food. Their responses were coded (0: do not like; 1: like; 2: indifferent) and recorded as follows:

a. Use an 80% confidence interval to estimate the proportion of consumers who like the snack food.

b. Provide a statistical interpretation for the confidence interval you constructed in part a.

Short Answer

Expert verified

a. The 80% confidence interval is0.2170,0.3830

b.One is 80% confident that the true proportion of consumers who like the snack food lies between the interval0.2170,0.3830

Step by step solution

01

Given information

A random sample of 50 consumers taste-tested a new snack food. Their responses were coded according to some criteria (0: do not like; 1: like; 2: indifferent). This has been presented in the table above.

02

(a) Calculation of 80% confidence interval of estimate the proportion of consumers who like the snack food.

The sample size is n= 50

15 of the data value are a 1, which indicates x=15

p=xn=1550=0.3

Here the confidence coefficient is 80%, hence 伪=20%.Now from the standard normal distribution table,z伪2=1.28

The margin of error

E=z伪2脳p^1-p^n=1.28脳0.3脳1-0.350=0.0830

Hence 80% lower limit:p^-E=0.3-0.0830=0.2170

80% upper limit:p^+E=0.3+0.0830=0.3830

The 80% confidence interval is0.2170,0.3830

03

(b) Interpretation of confidence interval constructed in part a

One is 80% confident that the true proportion of consumers who like snack food lies between the interval0.2170,0.3830

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Most popular questions from this chapter

Suppose you have selected a random sample of n = 5 measurements from a normal distribution. Compare the standard normal z-values with the corresponding t-values if you were forming the following confidence intervals.

a. 80% confidence interval

b. 90% confidence interval

c. 95% confidence interval

d. 98% confidence interval

e. 99% confidence interval

f. Use the table values you obtained in parts a鈥揺 to sketch the z- and t-distributions. What are the similarities and differences?

Suppose you wish to estimate the mean of a normal population

using a 95% confidence interval, and you know from prior information that蟽2鈮�1

a. To see the effect of the sample size on the width of the confidence interval, calculate the width of the confidence interval for n= 16, 25, 49, 100, and 400.

b. Plot the width as a function of sample size non graph paper. Connect the points by a smooth curve and note how the width decreases as nincreases.

Largest private companies. IPOs鈥攊nitial public offerings of stock鈥攃reate billions of dollars of new wealth for owners, managers, and employees of companies that were previously privately owned. Nevertheless, hundreds of large and thousands of small companies remain privately owned. The revenues of a random sample of 15 firms from Forbes 216 Largest Private Companies list are given in the table below

a. Describe the population from which the random sample was drawn.

b. Use a 98% confidence interval to estimate the mean revenue of the population of companies in question

c. Interpret your confidence interval in the context of the problem

d. What characteristic must the population possess to ensure the appropriateness of the estimation procedure used in part b?

e. Suppose Forbes reports that the true mean revenue of the 216 companies on the list is $5.0 billion. Is the claim believable?

It costs you \(10 to draw a sample of size n = 1 and measure the attribute of interest. You have a budget of \)1,500.

a. Do you have sufficient funds to estimate the population mean for the attribute of interest with a 95% confidence interval 5 units in width? Assume蟽=14.

b. If you used a 90% confidence level, would your answer to part a change? Explain.

Calculate the finite population correction factor for each

of the following situations:

a. n = 50, N = 2,000

b. n = 20, N = 100

c. n = 300, N = 1,500
See all solutions

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