91Ó°ÊÓ

To try to understand the reason for returned goods, the manager of a store examines the records on 40 products that were returned in the last year. Reasons were coded by 1 for "defective," 2 for "unsatisfactory," and 0 for all other reasons, with the results shown in the table. $$ \begin{array}{llllllllll} 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{array} $$ a. Give a point estimate of the proportion of all returns that are because of something wrong with the product, that is, either defective or performed unsatisfactorily. b. Assuming that the sample is sufficiently large, construct an \(80 \%\) confidence interval for the proportion of all returns that are because of something wrong with the product.

Step by step solution

01

Understanding the Point Estimate

To find the point estimate of the proportion of returns involving something wrong with the product, count how many instances of '1' (defective) and '2' (unsatisfactory) are recorded among the 40 entries. First, count the number of 1's and 2's.

02

Calculating the Point Estimate

The total number of 'defective' and 'unsatisfactory' records is 1 (defective) + 4 (unsatisfactory) = 5. The point estimate for the proportion is the number of these records divided by the total number of records. Thus, the point estimate is \( \hat{p} = \frac{5}{40} = 0.125 \).

03

Preparing for the Confidence Interval Calculation

Use the formula for the confidence interval for a proportion, \( \hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( n = 40 \) is the sample size and \( z \) is the z-score for an 80% confidence level.

04

Finding the Z-Score for 80% Confidence

For an 80% confidence interval, the z-score corresponds approximately to 1.28. This value can be found in standard normal distribution tables.

05

Calculating the Confidence Interval

Substitute \( \hat{p} = 0.125 \), \( n = 40 \), and \( z = 1.28 \) into the confidence interval formula. Calculate the margin of error: \(ME = 1.28 \times \sqrt{\frac{0.125 \times (1-0.125)}{40}}\).\This yields a margin of error of approximately 0.063. The confidence interval is thus \(0.125 \pm 0.063\), or from 0.062 to 0.188.

06

Interpreting the Results

The point estimate of the proportion of all returns due to defective or unsatisfactory products is 0.125. The 80% confidence interval for this proportion is (0.062, 0.188), reflecting some uncertainty due to sample size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Estimate

A point estimate provides a single value estimate for a statistical parameter, and in this context, it's used to represent the proportion of defective or unsatisfactory goods in the store's returns. This is a foundational concept because it gives a snapshot of your data's central tendency. In this exercise, to find the point estimate, one must identify the number of returns signaled as defective or unsatisfactory, coded as '1' or '2'. By adding these up (which in this case is 5), and dividing by the total number of returns examined (40), we determine the point estimate. Thus, the point estimate or sample proportion is calculated as \( \hat{p} = \frac{5}{40} = 0.125 \). This indicates that 12.5% of the returns are due to defective or unsatisfactory products.

Sample Size

Sample size refers to the number of observations or data points obtained in a survey or experiment. It's crucial in determining the reliability of a point estimate. In this exercise, the sample size is 40, representing the number of returned products examined. A larger sample size generally provides more reliable estimates because it can better approximate the true population parameter. The choice of sample size affects the width of the confidence interval and the z-score calculation that follows. For practical purposes, the sample size's adequacy must be ensured to apply the normal approximation method when constructing confidence intervals.

Proportion Calculation

Proportion calculation is vital to understanding what fraction of the population holds a certain characteristic. Here, we calculate the proportion of returned products that are either defective or unsatisfactory. The process involves identifying the relevant categories, tallying their occurrences, and then dividing by the total to get \( \hat{p} = \frac{5}{40} = 0.125 \). This calculation becomes important when you move on to constructing confidence intervals, as it's the base point from which margins of error are added and subtracted. Knowing the proportion helps to predict future returns' reasons accurately.

Z-Score

The z-score represents the number of standard deviations a data point is from the mean. In estimating confidence intervals, it's used to find the probability that a sample statistic falls within a certain range. For an 80% confidence interval, we find that approximately 10% of the distribution tails lie beyond the z-score, making the z-score about 1.28. This value is not random but obtained from standard statistical z-tables. The z-score helps calculate the margin of error. In this case, using the formula \( z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), with \( z = 1.28 \), we construct the confidence interval, giving us a complete picture of our data's variability around the point estimate.