Question

You may need to use the appropriate technology to answer this question. According to a research center, 9% 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 16 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 (C) Is the proportion of returns at the store in the given city significantly different from the returns for the country as a whole? Provide statistical support for your answer. Develop appropriate hypotheses such that rejection of H, will support the conclusion that the proportion of returns at the store in the given city is significantly different from the returns for the country as a whole. o Ho: P = 0.09 Haip 0.09 O Ho:p < 0.09 Hip 20.09 Ho:p> 0.09 Haip < 0.09 o Hp > 0.09 Haip s 0.09 O Hop $0.09 Haip > 0.09 Find the value of the test statistic. (Round your answer to two decimal places.) Find the p-value. (Round your answer to four decimal places.) p-value = At a = 0.01, what is your conclusion? Do not reject Ho. There is insufficient evidence to conclude that the return rate for the store in the given city is different than the country's national return rate. Reject Ho. There is insufficient evidence to conclude that the return rate for the store in the given city is different than the country's national return rate. Do not reject Ho. There is sufficient evidence to conclude that the return rate for the store in the given city is different than the country's national return rate. Reject Ho. There is sufficient evidence to conclude that the return rate for the store in the given city is different than the country's national return rate.

Answer 1

Given that :

n = 80

x = 16

alpha = 0.05

c = 95% = 0.95

a) Construct a point estimate of the proportion

p^ = x / n = 16 / 80 = 0.2

b) Construct a 95% confidence interval for the porportion

confidence level 1-alpha = 0.95

$z_{a/2}=1.96$

Confidence interval

$\hat{p}-z_{a/2}*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$=0.2-1.96*\sqrt{\frac{0.2(1-0.2)}{80}}$ = 0.1124

p(1-P) P + 2a/2* n
$=0.2+1.96*\sqrt{\frac{0.2(1-0.2)}{80}}$=0.2876

Confidence intervel ( 0.1124 To 0.2476 )

c) determine hypothesis

Ho : p = 0.09

Ha : p != 0.09

Test stastics :

$z=\frac{\hat{p}-po}{\sqrt{\frac{po(1-po)}{n}}} = \frac{0.2-0.09}{\sqrt{\frac{0.09(1-0.09)}{80}}}$

z = 3.54

p value is probability

P = P ( Z < -3.54 or Z > 3.54 ) = 2P ( Z < -3.54 ) = 0

if P-value smaller than the significance level alpha

hull hypotheses reject

Option D

reject Ho. there is sufficient evidence to cunclude that the return rate for the store in the given city in the difference