91Ó°ÊÓ

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the population means. b. Test at a 2.5\% significance level whether the claim of the management of the New Century Bank is truc. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=01 ?\) What if \(\alpha=05\) ?

Step by step solution

01

- Formulate the Hypotheses

The first step in hypothesis testing is to specify null hypothesis (\(H_{0}\)) and alternative Hypothesis (\(H_{1}\)). The null hypothesis is : \(H_{0}: \mu_{1} = \mu_{2}\), meaning the population mean waiting time at The New Century Bank is the same as at the Public Bank. The alternative hypothesis is : \(H_{1}: \mu_{1} < \mu_{2}\), Meaning the population mean waiting time at The New Century Bank is less than at the Public Bank.

02

- Compute the Confidence Interval

The confidence interval for the difference between the two population means is given by: \(\mu_{1} - \mu_{2} \pm Z_{\alpha/2} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\). For a 97% confidence interval, \(\alpha = 0.03\) and thus \(Z\) value corresponding to \(\alpha/2\) or 0.015 in the Standard Normal Table is approximately -2.17. Substituting the given values: \(4.5 - 4.75 \pm -2.17 \sqrt{\frac{1.2^{2}}{200} + \frac{1.5^{2}}{300}}\).

03

- Conduct the Hypothesis Test

We will conduct this test at a 2.5% significance level. This means \(\alpha = 0.025\). Calculate the test statistic using the formula \(Z = \frac{(\bar{x}_{1}-\bar{x}_{2}) - (\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}}\). Since \(\mu_{1} - \mu_{2} = 0\) under the null Hypothesis, the formula simplifies to \(Z = \frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}}\). Substitute the given values into the formula to find the test statistic.

04

- Calculate the p-value

The p-value is the smallest significance level at which we would reject the null hypothesis. It can be found by comparing our test statistic value to a Z-distribution. Use a statistical table or a calculator to find the p-value corresponding to our test statistic.

05

- Making Decisions Based on the p-value

If the calculated p-value is less than \(\alpha = 0.01\) or \(\alpha = 0.05\), then null hypothesis will be rejected, and it will be concluded that the population mean waiting time at The New Century Bank is less than at the Public Bank. If p-value is not less than given \(\alpha\), then we would not reject the null hypothesis, meaning there is no significant evidence that the mean waiting time at The New Century Bank is less than at Public Bank.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval

A confidence interval is a range of values that is likely to contain the true difference between the population means. In this exercise, the confidence interval helps determine the range where the real difference in waiting times between New Century Bank and Public Bank lies. Calculating a confidence interval allows us to make probabilistic statements about the data.
To compute the confidence interval, we use the formula: \[\mu_{1} - \mu_{2} \pm Z_{\alpha/2} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\]Here:

• \(\mu_1\) and \(\mu_2\) are the population means for New Century Bank and Public Bank, respectively.
• \(Z_{\alpha/2}\) is the Z-score that corresponds to our confidence level, in this case, 97%.
• \(\sigma_1\) and \(\sigma_2\) are the standard deviations for the banks.
• \(n_1\) and \(n_2\) are the sample sizes.

For a 97% confidence interval, the Z-score value is approximately -2.17. By substituting these values, you find the interval within which the true mean difference likely falls.

Null Hypothesis

The null hypothesis is a starting assumption in hypothesis testing. It posits that there is no effect or difference, serving as a null point for comparison. Here, the null hypothesis (\(H_{0}\)), is "the mean waiting time at New Century Bank is equal to the mean waiting time at Public Bank" (\(\mu_{1} = \mu_{2}\)).
The alternative hypothesis (\(H_{1}\)) asserts that "the mean waiting time at New Century Bank is less than that at Public Bank" (\(\mu_{1} < \mu_{2}\)).
The goal is to determine whether there is enough statistical evidence to reject the null hypothesis. If the evidence suggests the null hypothesis is unlikely, it gets rejected in favor of the alternative hypothesis, indicating that New Century Bank offers a shorter mean waiting time than Public Bank.

p-value

The p-value is a critical component in hypothesis testing. It's a measure that helps us understand the probability of observing our data, or something more extreme, assuming the null hypothesis is true. A low p-value indicates that the observed data is quite unlikely under the null hypothesis.
In this exercise, calculating the p-value involves using the test statistic derived from the sample data and comparing it to a standard normal distribution. The smaller the p-value, the stronger the evidence against the null hypothesis.
If the p-value is less than the significance level (often called \(\alpha\)), we reject the null hypothesis. For example, if the p-value is less than 0.01 or 0.05, it suggests that the waiting times at New Century Bank are significantly shorter than those at Public Bank.

Significance Level

The significance level, often denoted as \(\alpha\), is the threshold used to decide whether to reject the null hypothesis. It indicates the probability of committing a Type I error, which occurs when a true null hypothesis is incorrectly rejected.
Common significance levels are 0.05 (5%) and 0.01 (1%). In this scenario, a 2.5% significance level (\(\alpha = 0.025\)) was chosen to test the hypothesis that New Century Bank has shorter wait times.
Choosing the right significance level balances the risk of a false positive (rejecting a true null hypothesis) and the power of the test (the ability to detect a true difference when one exists). With a 2.5% significance level, we have more stringent criteria for rejecting the null hypothesis, increasing confidence in our conclusion if the null hypothesis is rejected.