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Chapter 6: Q63E (page 363)

The following is a 90% confidence interval for p:(0.26, 0.54). How large was the sample used to construct thisinterval?

Short Answer

Expert verified

The required sample size is 34.

Step by step solution

01

Given information

The following is a 90% confidence interval for p is(0.26, 0.54)

02

Finding the sample size

The formula for the confidence interval for a population proportion is

(p^鈭�z伪/2p^(1鈭�p^)n,p^+z伪/2p^(1鈭�p^)n)

Here the confidence interval is (0.26, 0.54)

Therefore,p^鈭�z伪/2p^(1鈭�p^)n=0.26.....(i)p^+z伪/2p^(1鈭�p^)n=0.54.....(ii)

Adding the equation(i) and equation(ii)

2p^=0.26+0.54p^=0.802p^=0.40

The critical value for 90% confidence interval isz伪/2=z0.10/2=z0.05=1.645

Now putting the value ofp^ and z伪/2in equation(i) ,

role="math" localid="1658294262963" 0.40鈭�1.6450.40(1鈭�0.40)n=0.261.6450.40脳0.60n=0.140.40脳0.60n=(0.141.645)2n=0.24(0.141.645)2n=33.135n鈮�34

The required sample size is 34

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Most popular questions from this chapter

In each of the following instances, determine whether you would use a z- or t-statistic (or neither) to form a 90%confidence interval and then state the appropriate z- ort-statistic value for the confidence interval.

a. Random sample of size n = 32 from a normal distribution with a population mean of 60 and population standard deviation of 4.

b. Random sample of size n = 108 from an unknown population.

c. Random sample of size n = 12 from a normal distribution witha sample mean of 83 and sample standard deviation of 2.

d. Random sample of size n = 24 from a normal distribution withan unknown mean and sample standard deviation of 3.

The following random sample was selected from a normal distribution: 4, 6, 3, 5, 9, and 3.

  1. Construct a 90% confidence interval for the population mean
  2. Construct a 95% confidence interval for the population mean
  3. Construct a 99% confidence interval for the population mean
  4. Assume that the sample means x and sample standard deviation s remain the same as those you just calculated but are based on a sample of n = 25 observations rather than n = 6 observations. Repeat parts a鈥揷. What is the effect of increasing the sample size on the width of the confidence intervals?

FindZ伪/2for each of the following:

a.= .10

b.= .01

c.= .05

d.= .20

Oil content of fried sweet potato chips. The characteristics of sweet potato chips fried at different temperatures were investigated in the Journal of Food Engineering (September 2013). A sample of 6 sweet potato slices was fried at 130掳 using a vacuum fryer. One characteristic of interest to the researchers was internal oil content (measured in millions of grams). The results were: x炉=178and s=11. The researchers are interested in estimating the variance of the interval oil content measurements for sweet potato chips.

a. Identify the target parameter, in symbols and words.

b. Compute a 95% confidence interval for 蟽2.

c. What does it mean to say that the target parameter lies within the interval with 鈥�95% confidence鈥�?

d. What assumption about the data must be satisfied in order for the confidence interval to be valid?

e. To obtain a practical interpretation of the interval, part b, explain why a confidence interval for the standard deviation, 蟽, is desired.

f. Use the results, part b, to compute a 95% confidence interval for蟽 . Give a practical interpretation of the interval.

Suppose you wish to estimate a population mean correct to within .20 with probability equal to .90. You do not know 蟽2, but you know that the observations will range in value between 30 and 34.

a. Find the approximate sample size that will produce the desired accuracy of the estimate. You wish to be conservative to ensure that the sample size will be ample to achieve the desired accuracy of the estimate. [Hint: Using your knowledge of data variation from Section 2.6, assume that the range of the observations will equal 4蟽.]

b. Calculate the approximate sample size, making the less conservative assumption that the range of the observations is equal to 6蟽.
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