91Ó°ÊÓ

A local college cafeteria has a self-service soft ice cream machine. The cafeteria provides bowls that can hold up to 16 ounces of ice cream. The food service manager is interested in comparing the average amount of ice cream dispensed by male students to the average amount dispensed by female students. A measurement device was placed on the ice cream machine to determine the amounts dispensed. Random samples of 85 male and 78 female students who got ice cream were selected. The sample averages were \(7.23\) and \(6.49\) ounces for the male and female students, respectively. Assume that the population standard deviations are \(1.22\) and \(1.17\) ounces, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of ice cream amounts dispensed by all male and all female students at this college, respectively. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(1 \%\) significance level, can you conclude that the average amount of ice cream dispensed by all male college students is larger than the average amount dispensed by all female college students? Use both approaches to make this test.

Step by step solution

01

Calculate the point estimate

The point estimate of \( \mu_{1} - \mu_{2} \) is the difference between the sample means. Therefore, perform the subtraction: \(7.23 - 6.49\)

02

Construct a 95% Confidence Interval

The formula for a confidence interval is \( \bar{X}_{1} - \bar{X}_{2} \pm Z_{\alpha/2} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\). Plug in the given values: \(7.23 - 6.49 \pm 1.96 \sqrt{\frac{1.22^{2}}{85} + \frac{1.17^{2}}{78}} \)

03

Perform A Hypothesis Test

For a two-tailed test with \(\alpha = 0.01\), the null hypothesis is \(H_{0}: \mu_{1} = \mu_{2}\) and the alternative hypothesis is \(H_{1}: \mu_{1} \neq \mu_{2}\). The test statistic calculated using the sample data is \(Z = \frac{\bar{X}_{1} - \bar{X}_{2}}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}} = \frac{7.23 - 6.49}{\sqrt{\frac{1.22^{2}}{85} + \frac{1.17^{2}}{78}}}\). Compare the calculated Z score with the critical Z score for a 1% significance level (2.58).

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

In statistics, a confidence interval is a range of values that is used to estimate a population parameter. It provides an idea about how uncertain we are about our estimate. For example, if we say we have a 95% confidence interval, it means we are 95% confident that the true difference between the population means falls within that range.
To construct a 95% confidence interval, we use the formula:

• \[ \bar{X}_{1} - \bar{X}_{2} \pm Z_{\alpha/2} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}} \]

Here, \( \bar{X}_1\) and \(\bar{X}_2\) are the sample averages for the two groups, and \(\sigma_1\) and \(\sigma_2\) are the population standard deviations.
The term \( Z_{\alpha/2} \) is the Z-score that corresponds to your confidence level; for a 95% confidence interval, \( Z_{\alpha/2} \) is 1.96.
The confidence interval tells you not only about the difference in means but also about the variability of that difference. If this interval does not include zero, it suggests a significant difference between the groups.

Population Means

The concept of population means relates to the average value of a characteristic across an entire population. In our exercise,

• \(\mu_{1}\) is the population mean amount of ice cream dispensed by male students.
• \(\mu_{2}\) is the population mean amount dispensed by female students.

Under hypothesis testing, we try to make educated guesses about these values based on our sample data.
Understanding population means is crucial because it allows policymakers, researchers, and managers to make informed decisions based on what is happening on average within a group. This is why our example asks for comparison: to infer whether there is a notable difference between male and female students on average.

Sample Averages

Sample averages, or sample means, are calculated by summing all measured values and dividing by the total number of observations in the sample. They are practical estimates of population means when collecting data from every member of the population isn't feasible.
In our exercise:

• The sample average amount of ice cream dispensed by male students is 7.23 ounces.
• For female students, it is 6.49 ounces.

The difference between these sample averages (7.23 - 6.49) gives us a point estimate of how much more, on average, male students might dispense than female students.

Significance Level

The significance level (\(\alpha\)) in hypothesis testing is the probability of rejecting the null hypothesis when it is actually true. It is often set at 0.05 (5%) for many tests, but for more strict tests, it might be 0.01 (1%) as seen in our exercise.
In hypothesis terms,

• The null hypothesis (\( H_{0} \)): Assumes no difference exists, \( \mu_{1} = \mu_{2} \).
• The alternative hypothesis (\( H_{1} \)): Posits a difference, \( \mu_{1} eq \mu_{2} \).

By comparing the calculated test statistic to a critical value (from a Z-distribution at the desired significance level), we determine whether to accept or reject the null hypothesis.
A 1% significance level means we are allowing a 1% chance of concluding that there is a difference when there isn't one. This tightens our tests, reducing the chances of making a type I error.