Question

The paper referenced in the previous exercise also gave information on calorie content. For the sample of Burger King meal purchases, the mean number of calories was 1,008 , and the standard deviation was \(483 .\) For the sample of McDonald's meal purchases, the mean number of calories was 908 , and the standard deviation was 624 . Based on these samples, is there convincing evidence that the mean number of calories in McDonald's meal purchases is less than the mean number of calories in Burger King meal purchases? Use \(\alpha=0.01\).

Short answer

The statistical decision will depend on the calculated t-value from step 2 and the rejection region from step 3. Without having the exact sample sizes, this decision cannot be given.

Step-by-step solution

State the Hypotheses

The Null Hypothesis (\(H_0\)) will be that the mean number of calories in McDonald's meal purchases is equal to or greater than that of Burger King purchases. The Alternative Hypothesis (\(H_1\)) will be that the mean number of calories in McDonald's meal purchases is less than that of Burger King purchases. In mathematical terms, \n Null Hypothesis - \(H_0: \mu_{McDonald} \geq \mu_{BurgerKing}\) \n Alternative Hypothesis - \(H_1: \mu_{McDonald} < \mu_{BurgerKing}\)

Calculate the Test Statistic

We use this formula to calculate the test statistic for the two-sample t-test:\n \[ t = \frac {(\bar{x}_1 - \bar{x}_2) - (D_0)} {\sqrt { \frac {s^2_1} {n_1} + \frac {s^2_2} {n_2} } } \] Where: \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(D_0\) is the hypothesized difference between population means, here D_0 = 0, because assuming that both means are equal under the null hypothesis, \(s^2_1\) and \(s^2_2\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes. The given information will be plugged into this formula to obtain the t-value.

Determine the Rejection Region

For a test of significance level \(\alpha=0.01\), since this is a left-tailed test (as we're checking if McDonald's mean is less than Burger King's), we find the critical value from t-distribution table with suitable degrees of freedom. If the calculated t-value is less than the critical value, we reject the null hypothesis.

Make the Statistical Decision

If the calculated t-value from step 2 falls in the Rejection Region determined in step 3, the Null hypothesis is rejected and it is concluded that there is convincing evidence that the mean number of calories in McDonald's meal purchases is less than the mean number of calories in Burger King meal purchases.

Key concepts

Two-Sample T-Test

The two-sample t-test is a statistical method used to determine whether there is a significant difference between the means of two independent samples. This type of analysis is helpful when comparing groups to see if they come from the same population with regard to the variable being measured.

For example, when comparing the calorie content of meals from two different fast-food chains, a two-sample t-test allows us to evaluate if the observed differences in calories are statistically significant or if they could have occurred by random chance. The t-test uses the means, standard deviations, and sample sizes of both groups to calculate a t-value. Comparing this t-value with a critical value from the t-distribution (considering the desired significance level and the degrees of freedom), we can decide whether to accept or reject the null hypothesis.

It is important to ensure that the samples are independent of each other and that the data is approximately normally distributed. The test can be two-tailed or one-tailed depending on the alternative hypothesis - in the described exercise, it is a one-tailed test because we are only interested in whether the mean of one group is less than the other.

Standard Deviation

Standard deviation is a measure of how spread out the values in a data set are around the mean, and it's critical in hypothesis testing to assess variability within each group. A small standard deviation means that the data points are clustered closely around the mean, while a large standard deviation indicates that the data points are spread out over a wider range.

In our exercise, the standard deviations of the meal calories from Burger King and McDonald's are given as 483 and 624 respectively. These values are essential for calculating the standard error of the difference between the two means, which in turn is used to compute the t-value for the two-sample t-test. The standard deviation plays a pivotal role in determining how much we expect our mean to vary if we were to take multiple samples from the population. Understanding standard deviation hence helps us to interpret the results of the t-test and assess the reliability of our data.

Null and Alternative Hypotheses

In the realm of hypothesis testing, the null hypothesis ( .) states that there is no difference in the variable being tested across groups, and serves as a skeptical assumption that any observed differences are due to random chance. The alternative hypothesis ( .), on the other hand, is the supposition that there is indeed a difference, and the direction of this difference is specified relative to the null hypothesis.

In our scenario, the null hypothesis assumes that there is no difference in calorie content between meals from McDonald's and Burger King, or that McDonald's might even have higher or equal calories. The alternative hypothesis posits that McDonald's meals have fewer calories than Burger King's. The hypothesis test then proceeds to investigate whether the data from the samples support the null hypothesis or if there's sufficient evidence to accept the alternative hypothesis. Establishing these hypotheses before conducting the analysis is a cornerstone of the scientific method and ensures objective testing.