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The paper "The Truth About Lying in Online Dating Profiles" (Proceedings, Computer-Human Interactions [2007]\(: 1-4)\) describes an investigation in which 40 men and 40 women with online dating profiles agreed to participate in a study. Each participant's height (in inches) was measured and the actual height was compared to the height given in that person's online profile. The differences between the online profile height and the actual height (profile - actual) were used to compute the values in the accompanying table. $$ \begin{array}{ll} \text { Men } & \text { Women } \\ \hline \bar{x}_{d}=0.57 & \bar{x}_{d}=0.03 \\ s_{d}=0.81 & s_{d}=0.75 \\ n=40 & n=40 \end{array} $$ For purposes of this exercise, assume it is reasonable to regard the two samples in this study as being representative of male online daters and female online daters. (Although the authors of the paper believed that their samples were representative of these populations, participants were volunteers recruited through newspaper advertisements, so we should be a bit hesitant to generalize results to all online daters!) a. Use the paired \(t\) test to determine if there is convincing evidence that, on average, male online daters overstate their height in online dating profiles. Use \(\alpha=.05\) b. Construct and interpret a \(95 \%\) confidence interval for the difference between the mean online dating profile height and mean actual height for female online daters. c. Use the two-sample \(t\) test of Section 11.1 to test \(H_{0}: \mu_{m}-\mu_{f}=0\) versus \(H_{a}: \mu_{m}-\mu_{f}>0,\) where \(\mu_{m}\) is the mean height difference (profile - actual) for male online daters and \(\mu_{f}\) is the mean height difference (profile - actual) for female online daters. d. Explain why a paired \(t\) test was used in Part (a) but a two-sample \(t\) test was used in Part (c).

Step by step solution

01

Paired t-test for Male Online Daters

The null hypothesis is that the mean difference between the profile and actual height for male daters is zero and the alternate hypothesis is that it is greater than zero. A paired t-test is a statistical procedure to determine if there's a significant difference between the means of two paired sets of data. Here, the sets of data are the actual heights and profile heights of the males. Using the provided values, the t statistic can be calculated using the formula \((\bar{x}_{d} - \mu)/(s_d/\sqrt{n})\), where \(\bar{x}_{d}\) is the mean difference, \(\mu\) is the expected mean difference (0), \(s_d\) is the standard deviation and \(n\) is the sample size. If the calculated t stat is greater than the critical t value at \(\alpha = 0.05\) for \(df = n - 1 = 39\) degrees of freedom, the null hypothesis is rejected.

02

Confidence Interval for Female Online Daters

The 95% confidence interval for the difference between the profile and actual height for female daters can be calculated by \(\bar{x}_{d} \pm (t_{critical} * s_d/\sqrt{n})\), where \(t_{critical}\) is the t value from the t-distribution table with \(df = n - 1\) corresponding to the desired confidence level (95%). This interval gives a range of values for the mean difference, within which the population mean difference is expected to lie with 95% confidence.

03

Two-Sample t-test for Difference in Mean Height

A two-sample t-test is used to check if there's a difference in means between two independent populations. Here, we want to test if there's a significant difference between the mean heights of the male and female online daters. The null hypothesis is that the mean height difference (profile - actual) is the same for both sexes, and the alternative hypothesis is that males have a greater difference. The test statistic can be calculated by using the formula \( t = (\bar{x}_{m} - \bar{x}_{f}) / \sqrt{((s_{m}^2/n_m) + (s_{f}^2/n_f))}\). If the calculated t stat is greater than the critical t value at \(\alpha = 0.05\) for \(df = n_m + n_f - 2\), the null hypothesis is rejected.

04

Explanation of Test Selection

A paired t-test is conducted in part a) because it's checking the difference between actual and profile heights within the same group of individuals (males), hence they are 'paired' measurements. In part c), a two-sample t-test is suitable because it's comparing the mean differences between two independent groups (males and females), hence the measurements are not paired.

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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 used to estimate an unknown population parameter. It's an interval calculated from your sample data that believes the true parameter value falls within this range, with a specified level of confidence. In the exercise, we constructed a 95% confidence interval for the difference between the actual and profile height of female online daters.

This means that we're 95% confident that the true mean difference in heights for all female online daters falls within our calculated range.

• The formula utilized is \( \bar{x}_{d} \pm (t_{critical} \times s_d/\sqrt{n}) \)
• \( \bar{x}_{d} \) is the sample mean difference.
• \( t_{critical} \) is a value derived from the t-distribution that adjusts as per our confidence level and degrees of freedom.

It's important to remember that the interval does not provide an exact value but instead gives insight into the range where the true difference likely lies.

Two-sample t-test

The two-sample t-test is a statistical method used to determine if two independent samples have different means. In our exercise, this was employed to verify whether the mean height difference for male online daters was greater than that for female online daters.

The critical part of the two-sample t-test is that each group (males and females) is independent of the other. This means that the data from one group doesn't affect the data from the other, which is not the case in the paired t-test setup.

• The null hypothesis is that both groups have equal mean height differences: \( H_0: \mu_{m} - \mu_{f} = 0 \)
• The alternative hypothesis is that the mean difference for males is greater: \( H_a: \mu_{m} - \mu_{f} > 0 \)

If the test statistic, calculated as \((\bar{x}_{m} - \bar{x}_{f}) / \sqrt{((s_{m}^2/n_m) + (s_{f}^2/n_f))}\), is larger than the critical value, it indicates that the null hypothesis can be rejected, suggesting a statistically significant difference in means between the groups.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It's a statement suggesting that there is no effect or no difference, and acts as a starting point for statistical comparison. In this exercise:

• For part (a), the null hypothesis asserts that there is no difference in the mean height difference for male online daters: \( H_0: \mu_d = 0 \).
• For part (c), the null posits that the mean difference between males and females is equal: \( H_0: \mu_m - \mu_f = 0 \).

When conducting a test, if the null hypothesis is not rejected, it means that there's not enough evidence to claim a significant difference. Conversely, if it is rejected, it means that the observed data provide strong evidence for an effect or a difference. It's essential in guiding the direction and outcome of the test, serving as a baseline comparison.

Statistical Significance

Statistical significance helps us determine the strength of the evidence against the null hypothesis. It's a measure of whether our observed data would likely happen under the assumption that the null hypothesis is true. Significance is often expressed with a significance level, denoted as \( \alpha \), commonly set at 0.05.

In the context of the exercise:

• For the paired t-test, a result is statistically significant if the observed t-statistic is greater than the critical t value for \( \alpha = 0.05 \), indicating that we can reject the null hypothesis that males do not overstate height.
• For the two-sample t-test, statistical significance suggests that the difference in the mean height differences between men and women is indeed real (as opposed to occurring by random chance).

Reaching statistical significance is crucial for validating our results. It tells us that our findings are not merely due to random variability, but that there's a genuine underlying effect or difference.