Question

Effect of Splitting the Bill Exercise 2.153 on page 105 describes a study to compare the cost of restaurant meals when people pay individually versus splitting the bill as a group. In the experiment half of the people were told they would each be responsible for individual meal costs and the other half were told the cost would be split equally among the six people at the table. The 24 people paying individually had a mean cost of 37.29 Israeli shekels with a standard deviation of 12.54 , while the 24 people splitting the bill had a higher mean cost of 50.92 Israeli shekels with a standard deviation of 14.33. The raw data can be found in SplitBill and both distributions are reasonably bell-shaped. Use this information to find and interpret a \(95 \%\) confidence interval for the difference in mean meal cost between these two situations.

Short answer

Without performing the actual calculations, the steps detailed above should lead to the 95% confidence interval for the difference in mean meal cost. The presence or absence of zero in this interval will denote whether a significant difference between the two situations exists.

Step-by-step solution

Formulate the Confidence Interval Formula

The formula for a confidence interval around a difference in means is given by \((\overline{X}_1 - \overline{X}_2) \pm Z \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\), where \(\overline{X}_i\) is the mean, \(s_i\) the standard deviation, \(n_i\) the sample size for group i, and Z the z-score associated with our confidence level. For a 95% confidence level, Z is approximately 1.96.

Compute the Confidence Interval

Plugging in the given values in the formula: \(\overline{X}_1 = 37.29\), \(s_1 = 12.54\), \(n_1 = 24\), \(\overline{X}_2 = 50.92\), \(s_2 = 14.33\), \(n_2 = 24\), and \(Z = 1.96\), we can calculate the confidence interval: \(CI = (37.29 - 50.92) \pm 1.96 \cdot \sqrt{\frac{12.54^2}{24} + \frac{14.33^2}{24}}\).

Interpret the Confidence Interval

Once calculated, the confidence interval represents the range of plausible values for the difference in mean meal cost between the two situations. If the interval contains zero, it would indicate that there is no significant difference between the two means at the 95% confidence level. If the interval does not contain zero, it indicates that a significant difference exists.