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When conducting a study to compare the costs of restaurant meals based on how bills are paid, examining the difference in means is crucial. The mean is essentially the average cost in each group, and the difference in means helps us to assess whether there is a statistically significant difference between two setups: paying individually or splitting the bill equally.
In this exercise, the people who paid individually had a mean cost of 37.29 shekels, while the group splitting the bill had a mean of 50.92 shekels. The difference between these two means is one of the primary values of interest in this experiment.
By subtracting the individual payment mean from the group payment mean, we find the difference in means: \[\overline{X}_1 - \overline{X}_2 = 37.29 - 50.92 = -13.63\]This negative difference indicates that, on average, those who paid individually spent less compared to those who split the bill, highlighting an important trend in how payment methods can influence spending behavior.

Understanding standard deviation is key to interpreting the variability of data in each group. Standard deviation tells us how spread out the meal costs are around their respective means in both scenarios.
In our exercise: - For those paying individually, the standard deviation is 12.54 shekels. - For those splitting the bill, the standard deviation is 14.33 shekels.
These values indicate the degree of variation in meal costs. A lower standard deviation implies that meal costs are closer to the mean, whereas a higher standard deviation suggests a wider range of meal costs.
These standard deviations are incorporated into further calculations, such as the confidence interval, as they affect the precision of our estimates regarding the mean differences. A larger standard deviation can lead to a wider confidence interval, reflecting greater uncertainty about the precise difference in means.

A 95% confidence level is a popular statistical threshold used to estimate the range of values that is likely to contain the true difference in means. When we use this confidence level, we are saying that if we were to repeat this study many times, 95% of the time the calculated confidence interval would encompass the true mean difference.
The formula for our confidence interval around the difference in means includes a crucial term: \[CI = (\overline{X}_1 - \overline{X}_2) \pm Z \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]Where \(Z\) is the z-score that corresponds to the chosen confidence level, here approximately 1.96 for 95%.
Once we compute the confidence interval, it provides a range of plausible values for the difference in means. Importantly, if zero is not within this interval, we can say with 95% confidence that a significant difference exists between the two bill scenarios. If zero lies within the interval, it indicates that there might not be a significant difference in spending behavior between the two payment methods.