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Happy customers As the Hispanic population in the United States has grown, businesses have tried to understand what Hispanics like. One study interviewed a random sample of customers leaving a bank. Customers were classified as Hispanic if they preferred to be interviewed in Spanish or as Anglo if they preferred English. Each customer rated the importance of several aspects of bank service on a 10-point scale.\(^{34}\) Here are summary results for the importance of 鈥渞eliability鈥� (the accuracy of account records and so on): $$\begin{array}{llll}{\text { Group }} & {n} & {\overline{x}} & {s_{x}} \\\ \hline \text { Anglo } & {92} & {6.37} & {0.60} \\ {\text { Hispanic }} & {86} & {5.91} & {0.93}\end{array}$$ (a) The distribution of reliability ratings in each group is not Normal. A graph of the data reveals no outliers. The use of two-sample t procedures is justified. Why? (b) Construct and interpret a 95% confidence interval for the difference between the mean ratings of the importance of reliability for Anglo and Hispanic bank customers. (c) Interpret the 95% confidence level in the context of this study.

Step by step solution

01

Justification for Two-Sample T-Test

Though the distribution of reliability ratings is not Normal, the sample sizes for both groups (Anglo: 92 and Hispanic: 86) are large enough to use the Central Limit Theorem, which allows us to use two-sample t procedures. The Central Limit Theorem suggests that with large sample sizes, the sampling distribution of the sample mean approaches a Normal distribution regardless of the shape of the population distribution. Also, since there are no outliers, this further justifies using the two-sample t-test.

02

Calculate Standard Error of the Difference in Means

To construct a confidence interval, we first need the standard error (SE) of the difference between the sample means. The formula for SE is given by: \[ \text{SE} = \sqrt{\frac{s_{x_1}^2}{n_1} + \frac{s_{x_2}^2}{n_2}} \] Using this formula: \[ \text{SE} = \sqrt{\frac{0.60^2}{92} + \frac{0.93^2}{86}} \approx 0.141 \]

03

Determine the Margin of Error

The margin of error (ME) for the confidence interval is calculated using the critical t-value at 95% confidence level and the SE. For large samples, we use a z-score close to t-value, which is approximately 1.96. Thus: \[ \text{ME} = 1.96 \times 0.141 \approx 0.276 \]

04

Construct the 95% Confidence Interval

Now, use the sample means and margin of error to construct the confidence interval for the difference in means \((\overline{x}_1 - \overline{x}_2)\): \[ (6.37 - 5.91) \pm 0.276 \] \[ 0.46 \pm 0.276 \] \[ [0.184, 0.736] \] This means the confidence interval for the difference between the mean ratings is (0.184, 0.736).

05

Interpret the Confidence Interval

The 95% confidence interval from Step 4 means that we can be 95% confident that the true difference in the mean ratings of reliability between Anglo and Hispanic customers lies between 0.184 and 0.736. This suggests that on average, Anglo customers rate reliability higher compared to Hispanic customers.

06

Interpret the Confidence Level

The 95% confidence level indicates that if we were to repeat this study many times, approximately 95% of the confidence intervals computed from those samples would capture the true difference in the mean ratings of reliability between Anglo and Hispanic bank customers.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem (CLT) is a cornerstone of statistics that makes it possible to use the t-test even when data distributions aren't normal. It tells us that regardless of the original distribution shape, the distribution of the sample mean will tend to be normal, provided the sample size is large enough. For this reason, the CLT justifies the use of two-sample t procedures in the absence of normality, as long as the sample size is sufficiently large. In our example, though the reliability ratings don't follow a normal distribution, the samples sizes (92 for Anglo and 86 for Hispanic) are large enough for the sampling distribution of their means to be approximated by a normal distribution. This is why using the two-sample t-test is appropriate in this scenario. No presence of outliers further supports this decision.

Confidence Interval

A confidence interval gives a range of values within which we believe the true parameter, like a population mean difference, lies. It provides a way to estimate the uncertainty of this parameter from sample data. In this exercise, the 95% confidence interval for the difference in mean ratings of reliability between Anglo and Hispanic customers was computed to be between 0.184 and 0.736. This range means that we can be 95% confident that the actual difference in mean ratings for the larger population of bank customers falls within these values. The interval calculation accounts for the variability in the sample data, providing a statistical measure of precision for our estimate.

Margin of Error

The margin of error plays a key role in constructing confidence intervals. It quantifies the extent to which the sample estimate varies from the true population value, based on the data's variability and sample size. In this scenario, the margin of error was calculated to be approximately 0.276, using a critical t-value approximated by a z-score of 1.96 (typical for 95% confidence levels) and the standard error. The margin of error encompasses the likely range of variability, allowing us to extend the confidence interval symmetrically from the sample mean difference. This padding around the sample estimate helps account for possible errors in the estimate due to randomness in sampling.

Sample Mean

The sample mean is a fundamental concept in statistics, serving as a measure of central tendency. It sums all observed values and divides by the number of observations, thus providing an average that represents the data set. In the context of this problem, the sample mean for the Anglo group was 6.37, while for the Hispanic group it was 5.91. These values serve as point estimates for the population means of each group. They are critical inputs for statistical calculations, like the confidence interval. When combined with the other components, such as sample size and standard deviation, they help in assessing and comparing the importance of bank service reliability between different customer segments.