Question

A clothing company produces denim jeans. The jeans are made and sold with either a regular cut or a boot-cut. To estimate the proportion of all customers in Tacoma, WA, who prefer boot-cut jeans, a marketing researcher examined sales receipts for a random sample of 178 customers who purchased jeans from the firm’s Tacoma store. 56 of the customers in the sample purchased boot-cut jeans. Construct the 99% confidence interval to estimate the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans and interpret the confidence interval (please write the interval boundaries to THREE decimal places

Answer 1

99% confidence interval for the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans is [0.225 , 0.405].

Explanation:

We are given that a marketing researcher examined sales receipts for a random sample of 178 customers who purchased jeans from the firm’s Tacoma store. 56 of the customers in the sample purchased boot-cut jeans.

Firstly, the Pivotal quantity for 99% confidence interval for the population proportion is given by;

P.Q. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of customers who purchased boot-cut jeans =
`(56)/(178)` = 0.315

n = sample of customers = 178

p = population proportion of customers who prefer boot-cut jeans

Here for constructing 99% confidence interval we have used One-sample z test for proportions.

So, 99% confidence interval for the population proportion, p is ;

P(-2.58 < N(0,1) < 2.58) = 0.99 {As the critical value of z at 0.5% level

of significance are -2.58 & 2.58}

P(-2.58 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 2.58) = 0.99

P(
`-2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

P(
`\hat p-2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

99% confidence interval for p = [
`\hat p-2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }`,
`\hat p+2.58 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.315-2.58 * {\sqrt{(0.315(1-0.315))/(178) } }` ,
`0.315+2.58 * {\sqrt{(0.315(1-0.315))/(178) } }` ]

= [0.225 , 0.405]

Therefore, 99% confidence interval for the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans is [0.225 , 0.405].

The interpretation of the above confidence interval is that we are 99% confident that the proportion of all customers in Tacoma, Washington, who prefer boot-cut jeans will lie between 0.225 and 0.405.