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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. \({ }^{31}\) Here are summary results for the importance of "reliability" (the accuracy of account records and so on $$ \begin{array}{lccc} \hline \text { Group } & n & \bar{x} & s_{x} \\ \text { Anglo } & 92 & 6.37 & 0.60 \\ \text { Hispanic } & 86 & 5.91 & 0.93 \\ \hline \end{array} $$ (a) The distribution of reliability ratings in each group is not Normal. The use of two-sample \(t\) procedures is still 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

Understand Normality Assumption

Two-sample t procedures are justified even if the distribution of the data is not exactly Normal, provided that the sample sizes are large enough. Generally, a sample size of 30 or more is considered sufficient because of the Central Limit Theorem, which suggests that the distribution of sample means tends to be Normal regardless of the shape of the population distribution.

02

Calculate the Standard Error

We calculate the standard error (SE) for the difference in means using the formula: \( SE = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}} \), where \( s_{1} \) and \( s_{2} \) are the standard deviations for the Anglo and Hispanic groups, and \( n_{1} \) and \( n_{2} \) are the sample sizes. Plugging in the values: \( SE = \sqrt{\frac{0.60^2}{92} + \frac{0.93^2}{86}} = \sqrt{\frac{0.36}{92} + \frac{0.8649}{86}} = \sqrt{0.003913 + 0.010053} \approx \sqrt{0.013966} \approx 0.1182 \)

03

Calculate the Confidence Interval

The formula for the confidence interval for the difference in means is: \( (\bar{x}_{1} - \bar{x}_{2}) \pm t^{*} \times SE \) where \( \bar{x}_{1} = 6.37 \) (Anglo mean) and \( \bar{x}_{2} = 5.91 \) (Hispanic mean), and \( t^{*} \) is the critical value for the 95% confidence level. For large samples, we can approximate \( t^{*} \approx 1.96 \). Difference in means \( = 6.37 - 5.91 = 0.46 \). Confidence Interval: \( 0.46 \pm 1.96 \times 0.1182 \approx 0.46 \pm 0.2317 \). Therefore, the 95% confidence interval is \( (0.2283, 0.6917) \).

04

Interpret the Confidence Interval

The 95% confidence interval for the difference in means, \( (0.2283, 0.6917) \), indicates that the true difference in mean ratings of reliability between Anglo and Hispanic customers is between 0.2283 and 0.6917. Since the interval does not include 0, there is a statistically significant difference in the importance of reliability between these two groups.

05

Interpret the Confidence Level

The 95% confidence level means that if we repeated this study numerous times, approximately 95% of the confidence intervals constructed from those samples would contain the true difference in mean ratings between Anglo and Hispanic customers. It gives us a measure of the reliability of our interval estimate.

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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 is a fundamental concept in statistics that plays a crucial role when working with non-Normal distributions. It states that when you have a large enough sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the original population distribution's shape.
This theorem is the reason why two-sample t-tests can still be valid even if the data itself isn't normally distributed. When the sample size is large, say more than 30, the mean of those samples tends to follow a normal distribution due to the Central Limit Theorem.

• Therefore, in studies like this one comparing Anglo and Hispanic customers, the sample size of 92 and 86 is adequate enough to satisfy the conditions of the theorem.
• It assures us that we can trust the results obtained from the t-test procedures.

This reliability stems from the notion that the sample means will be normally distributed, allowing you to use the known properties of normal distribution to make inferences.

Confidence Interval

A confidence interval provides a range of values, which is used to estimate the unknown parameter of the population, in this case, the difference between means. In simple terms, it answers the question of what range the true difference between Anglo and Hispanic customers' reliability ratings would fall under.
In our study, the difference in mean is computed as 0.46, and a 95% confidence interval is calculated using this difference. The interval, in your example, is from approximately 0.2283 to 0.6917.

• This means there is 95% confidence that the true average difference in ratings between the two groups lies within this interval.
• Since the interval doesn't include 0, it suggests there is a significant difference in perceptions of reliability between these groups.

Confidence intervals give us more than just a guess - they offer an estimate that includes variability and uncertainty, assuring we have captured the true parameter most of the time, given many samples.

Standard Error

The standard error measures the accuracy with which a sample represents a population. Specifically, in the context of a two-sample t-test, it does so for the difference between two sample means.
In the given exercise, the formula for the standard error of the difference in means is \[ SE = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}} \]where \( s_{1} \) and \( s_{2} \) are the standard deviations, and \( n_{1} \) and \( n_{2} \) are the sample sizes for the Anglo and Hispanic groups.

• This calculation, resulting in 0.1182, tells us how much we can expect the calculated sample mean difference to fluctuate if we were to repeat the experiment multiple times.
• It also plays a significant role in determining the width of the confidence interval – the smaller the standard error, the narrower and more precise the interval.

A smaller standard error signifies that your sample mean is a better reflection of the true population mean.

Normality Assumption

The normality assumption in statistical tests, such as the two-sample t-test, implies that the data are distributed in a bell-shaped curve, known as the normal distribution. However, in practice, data are seldom perfectly normal. Fortunately, the Central Limit Theorem allows flexibility here.
According to it, for sufficiently large samples (generally \( n \geq 30 \)), the distribution of the sample mean will still approximate a normal distribution even if the underlying data are not normal.

• This is why the two-sample t-test can still be valid in this study comparing customer ratings, despite the original distributions in each group not being perfectly normal.
• The assumption holds provided that the data aren't extremely skewed or the sample sizes aren't too small.

This gives researchers assurance that even with imperfect data, they can conduct reliable inferences about the differences between group means.