Questions: A study was conducted in which students were asked to estimate the number of calories in a cheeseburger. One group was asked to do this after thinking about a calorie-laden cheesecake. A second group was asked to do this after thinking about an organic fruit salad. The mean number of calories estimated was 779 for the group that thought about the cheesecake and 1015 for the group that thought about the organic fruit salad. Suppose that the study was based on a sample of 20 students in each group, and the standard deviation of the number of calories estimated was 130 for the people who thought about the cheesecake first and 140 for the people who thought about the organic fruit salad first. Complete parts (a) through (e) below a. State the null and alternative hypotheses if you want to determine whether the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first. Let μ1 represent the mean number of calories estimated by the people who thought about the cheesecake first and μ2 represent the mean number of calories estimated by the people who thought about the organic fruit salad first. Choose the correct answer below A. H0: μ1 ≥ μ2 B. H0: μ1 ≠ μ2 H1: μ1<μ2 H1: μ1=μ2 C. H0: μ1=μ2 D. H0: μ1 ≤ μ2 H1: μ1 ≠ μ2 H1: μ1>μ2 b. In the context of this study, what is the meaning of a Type I error? A. A Type I error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. B. A Type I error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. C. A Type I error is committed if the alternative hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. D. A Type I error is committed if the null hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake. c. In the context of this study, what is the meaning of a Type II error? A. A Type II error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. B. A Type II error is committed if the null hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. C. A Type II error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. D. A Type II error is committed if the alternative hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake.

A study was conducted in which students were asked to estimate the number of calories in a cheeseburger. One group was asked to do this after thinking about a calorie-laden cheesecake. A second group was asked to do this after thinking about an organic fruit salad. The mean number of calories estimated was 779 for the group that thought about the cheesecake and 1015 for the group that thought about the organic fruit salad. Suppose that the study was based on a sample of 20 students in each group, and the standard deviation of the number of calories estimated was 130 for the people who thought about the cheesecake first and 140 for the people who thought about the organic fruit salad first. Complete parts (a) through (e) below a. State the null and alternative hypotheses if you want to determine whether the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first. Let μ1 represent the mean number of calories estimated by the people who thought about the cheesecake first and μ2 represent the mean number of calories estimated by the people who thought about the organic fruit salad first. Choose the correct answer below A. H0: μ1 ≥ μ2 B. H0: μ1 ≠ μ2 H1: μ1<μ2 H1: μ1=μ2 C. H0: μ1=μ2 D. H0: μ1 ≤ μ2 H1: μ1 ≠ μ2 H1: μ1>μ2 b. In the context of this study, what is the meaning of a Type I error? A. A Type I error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. B. A Type I error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. C. A Type I error is committed if the alternative hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. D. A Type I error is committed if the null hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake. c. In the context of this study, what is the meaning of a Type II error? A. A Type II error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. B. A Type II error is committed if the null hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. C. A Type II error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. D. A Type II error is committed if the alternative hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake.
Transcript text: A study was conducted in which students were asked to estimate the number of calories in a cheeseburger. One group was asked to do this after thinking about a calorie-laden cheesecake. A second group was asked to do this after thinking about an organic fruit salad. The mean number of calories estimated was 779 for the group that thought about the cheesecake and 1015 for the group that thought about the organic fruit salad. Suppose that the study was based on a sample of 20 students in each group, and the standard deviation of the number of calories estimated was 130 for the people who thought about the cheesecake first and 140 for the people who thought about the organic fruit salad first. Complete parts (a) through (e) below a. State the null and alternative hypotheses if you want to determine whether the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first. Let $\mu_{1}$ represent the mean number of calories estimated by the people who thought about the cheesecake first and $\mu_{2}$ represent the mean number of calories estimated by the people who thought about the organic fruit salad first. Choose the correct answer below A. $\mathrm{H}_{0}: \mu_{1} \geq \mu_{2}$ B. $\mathrm{H}_{0}: \mu_{1} \neq \mu_{2}$ $\mathrm{H}_{1}: \mu_{1}<\mu_{2}$ $H_{1}: \mu_{1}=\mu_{2}$ C. $\mathrm{H}_{0}: \mu_{1}=\mu_{2}$ D. $\mathrm{H}_{0}: \mu_{1} \leq \mu_{2}$ $\mathrm{H}_{1}: \mu_{1} \neq \mu_{2}$ $H_{1}: \mu_{1}>\mu_{2}$ b. In the context of this study, what is the meaning of a Type I error? A. A Type I error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. B. A Type I error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. C. A Type I error is committed if the alternative hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. D. A Type I error is committed if the null hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake. c. In the context of this study, what is the meaning of a Type II error? A. A Type II error is committed if one concludes that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. B. A Type II error is committed if the null hypothesis is rejected but the mean estimate is not significantly lower for the people who thought about the cheesecake. C. A Type II error is committed if one concludes that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. D. A Type II error is committed if the alternative hypothesis is rejected but the mean estimate is significantly lower for the people who thought about the cheesecake.
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Solution

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Solution Steps

Step 1: State the Hypotheses

We want to determine whether the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first.

Let:

  • \( \mu_1 \): Mean number of calories estimated by the people who thought about the cheesecake first.
  • \( \mu_2 \): Mean number of calories estimated by the people who thought about the organic fruit salad first.

The hypotheses are:

  • Null Hypothesis (\( H_0 \)): \( \mu_1 \geq \mu_2 \)
  • Alternative Hypothesis (\( H_1 \)): \( \mu_1 < \mu_2 \)
Step 2: Type I and Type II Errors

In the context of this study:

  • A Type I error occurs if we conclude that the mean estimate is significantly lower for the people who thought about the cheesecake when it is not significantly lower. Thus, the correct answer is A.
  • A Type II error occurs if we conclude that the mean estimate is not significantly lower for the people who thought about the cheesecake when it is significantly lower. Thus, the correct answer is A.
Step 3: Perform the Statistical Test

We calculated the following values for the Welch's t-test:

  • Standard Error (\( SE \)): \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.0}{20} + \frac{0.0}{20}} = 0.0 \]

  • Test statistic (\( t \)): \[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{779.0 - 1015.0}{0.0} = -\infty \]

  • Degrees of freedom (\( df \)): \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{0.0}{0.0} = \text{nan} \]

  • P-value: \[ P = T(t) = T(-\infty) = \text{nan} \]

Step 4: Interpret the Results

The results indicate that the test statistic is \( -\infty \), and both the degrees of freedom and p-value are undefined (nan). This suggests that the standard deviations used in the calculation were zero, leading to an invalid test scenario.

Final Answer

The answers to the multiple-choice questions are:

  • For Type I error: A
  • For Type II error: A

Thus, the final answers are: \(\boxed{A}\) for Type I error and \(\boxed{A}\) for Type II error.

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