Question

According to a research center, 16% of all merchandise sold in a particular country gets returned. A department store in a certain city sampled 80 items sold in January and found that 24 of the items were returned. (a) Construct a point estimate of the proportion of items returned for the population of sales transactions at the store in the given city. (b) Construct a 95% confidence interval for the proportion of returns at the store in the given city. (Round your answers to four decimal places.) to

Answer 1

Answer:

a) point estimate of the proportion of items returned for the population of sales transactions at the store in the given city

Point estimate = 0.3

b) The 95% confidence interval for the proportion of returns at the store in the given city

( 0.19958,0.40042)

Step-by-step explanation:

Given the Population proportion 'P'= 16% = 0.16

Given data a department store in a certain city sampled 80 items sold in January and found that 24 of the items were returned

The sample proportion $p = \frac{x}{n}$

here x =24 and n = 80

The sample proportion $p = \frac{x}{n} = \frac{24}{80} = 0.3$

a) Point estimate:-

A point estimate of the true proportion P

Point estimate = $p = \frac{x}{n}$

point estimate of the proportion of items returned for the population of sales transactions at the store in the given city

$p = \frac{x}{n} = \frac{24}{80} = 0.3$

b) The 95%  confidence interval for the proportion

The 95% confidence interval for the proportion of returns at the store in the given city

$(p - z_{\alpha } \sqrt{\frac{p(1-p)}{n} } , p + z_{\alpha }\sqrt{\frac{p(1-p)}{n} } )$

substitute all values we get ,

$(0.3 - 1.96 \sqrt{\frac{0.3(1-0.3)}{80} } , 0.3 + 1.96\sqrt{\frac{0.3(1-0.3)}{80} } )$

on simplification, we get

(0.3-0.10042 ,0.3+0.10042)

( 0.19958,0.40042)

Conclusion:-

a)  point estimate of the proportion of items returned for the population of sales transactions at the store in the given city

Point estimate = 0.3

b) The 95% confidence interval for the proportion of returns at the store in the given city

( 0.19958,0.40042)