Aceording to the University of Nevada Center for Logistics...

Aceording to the University of Nevada Center for Logistics Management, $6 \%$ of all merchandise sold in the United Stares gets returned (BusinesrWeek, January 15, 2007). A Houston department store sampled 80 items sold in Janusry and found that 12 of the items were returned. a. Construct a point estimate of the proportion of items returned for the population of sales transactions at the Houston store. b. Construct a $95 \%$ confidence interval for the porportion of retums at the Houston store. c. Is the proportion of retums al the Houston store significantly different from the retums for the nation as a whole? Provide statistical support for your answer.

Aceording to the University of Nevada Center for Logistics Management, $6 \%$ of all merchandise sold in the United Stares gets returned (BusinesrWeek, January 15, 2007). A Houston department store sampled 80 items sold in Janusry and found that 12 of the items were returned.
a. Construct a point estimate of the proportion of items returned for the population of sales transactions at the Houston store.
b. Construct a $95 \%$ confidence interval for the porportion of retums at the Houston store.
c. Is the proportion of retums al the Houston store significantly different from the retums for the nation as a whole? Provide statistical support for your answer.

Key Concepts

Point Estimate

A point estimate is a single value derived from sample data used to estimate an unknown population parameter. In the context of proportions, the sample proportion, calculated as the number of observed successes divided by the total number of observations, serves as the point estimate for the true population proportion.

Confidence Interval for Proportions

A confidence interval for a proportion provides a range of plausible values for the population proportion, reflecting the uncertainty inherent in using a sample statistic as an estimate. It is constructed by taking the sample proportion and adding and subtracting a margin of error, which is calculated by multiplying the standard error of the proportion by the appropriate z-score for the desired confidence level.

Hypothesis Testing for Proportions

Hypothesis testing for proportions examines whether the observed sample proportion significantly differs from a known or established population proportion. This involves setting up a null hypothesis that the true proportion is equal to a specified value, computing a test statistic (typically a z-score) based on the difference between the sample and population proportions, and comparing this statistic to critical values to determine statistical significance.

Transcript

00:01 In this problem, we're given that our population proportion of merchandise sold in united states gets returned is 0 .06 from 6%.

00:13 And we are also given that our sample size is 80 items.

00:17 And the first part of the question asks us to compute a point statistic.

00:21 And in this situation, that would be a sample proportion, which is equal to the number of items returned in our sample over the number of items total in our sample.

00:31 Sample, which is equal to 0 .15.

00:35 So that is our sample proportion, and from this we also get that our sample size is 80.

00:43 So now for part b, we have to provide a 95 % confidence interval, and the formula for that is ci equals, ci stands for confidence interval.

01:01 The sample mean plus or minus the z value of our alpha over 2.

01:06 And our alpha is 1 minus 0 .95, 0 .05.

01:12 So our z score, in this case of 0 .025, times the square root of our sample mean times 1 minus our sample proportion, sorry, not sample mean, over our sample size.

01:26 That is our confidence interval.

01:30 And that is equal to our sample proportion in this situation is 0 .15 plus or minus.

01:40 A z test statistic of a 0 .5 over 2 level, so that is equal to 0 .025.

01:52 And if we go to our z table, we can find where our test statistic is 0 .025.

02:02 So if we plot out a z distribution to equal 0 is here, a probability of 0 .025 is over here.

02:15 So the probability equals 0 .025 is over here.

02:20 So what that means is we need to find a z value for this line right here.

02:27 In order to do that, we can take the area to the left of this z score and find that value in this plot somewhere.

02:37 So this probability here equal to 0 .975.

02:42 If we scan the chart to find .975, it's right here, and that is 1 .96.

02:51 So our z, i'm going to delete this, our z of 0 .025 is equal to 1 .96, 1 .96 times the square root of 0 .15 times 1 minus 0 .15 .85 over our sample size of 80.

03:15 And we get from that our confidence interval is 0 .0718 to 0 .2282.

03:28 This is the answer to part b.

03:33 So, in essence, we are, part b, we are 95 % confident that our true mean, or our true proportion, sorry, lies between 0 .0718 and 0 .2282.

04:11 And now we have to find a, or is there a sufficient, we have to find if there's a sufficient evidence, if there is sufficient evidence to claim a significant difference.

04:22 So our null hypothesis is that there is no difference...

Please give Ace some feedback