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Chapter 13: Problem 66

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

Expert verified
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

01

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 - \(H_1: \mu_{McDonald} < \mu_{BurgerKing}\)
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Babies born extremely prematurely run the risk of various neurological problems and tend to have lower IQ and verbal ability scores than babies that are not premature. The article "Premature Babies May Recover Intelligence, Study Says" (San Luis Obispo Tribune, February 12,2003 ) summarized medical research that suggests that the deficits observed at an early age may decrease as children age. Children who were born prematurely were given a test of verbal ability at age 3 and again at age 8 . The test is scaled so that a score of 100 would be average for normal-birth-weight children. Data for 50 children who were born prematurely were used to generate the accompanying Minitab output, where Age 3 represents the verbal ability score at age 3 and Age8 represents the verbal ability score at age \(8 .\) Use the Minitab output to determine if there is convincing evidence that the mean verbal ability score for children born prematurely increases between age 3 and age 8 . You can assume that it is reasonable to regard the sample of 50 children as a random sample from the population of all children born prematurely. $$ \begin{aligned} &\text { Paired T-Test and Cl: Age8, Age3 }\\\ &\begin{array}{l} \text { Paired } T \text { for } \text { Age8 - Age3 } \\ \begin{array}{rrrrr} & \text { N } & \text { Mean } & \text { StDev } & \text { Se Mean } \\ \text { Age8 } & 50 & 97.21 & 16.97 & 2.40 \\ \text { Age3 } & 50 & 87.30 & 13.84 & 1.96 \\ \text { Difference } & 50 & 9.91 & 22.11 & 3.13 \\ \text { T-Test of mean difference } & =0(\mathrm{vs}>0): \text { T-Value }=3.17 \\ \text { P-Value }=0.001 & & & \end{array} \end{array} \end{aligned} $$

Some people believe that talking on a cell phone while driving slows reaction time, increasing the risk of accidents. The study described in the paper "A Comparison of the Cell Phone Driver and the Drunk Driver" (Human Factors [2006]: \(381-391\) ) investigated the braking reaction time of people driving in a driving simulator. Drivers followed a pace car in the simulator, and when the pace car's brake lights came on, the drivers were supposed to step on the brake. The time between the pace car brake lights coming on and the driver stepping on the brake was measured. Two samples of 40 drivers participated in the study. The 40 people in one sample used a cell phone while driving. The 40 people in the second sample drank a mixture of orange juice and alcohol in an amount calculated to achieve a blood alcohol level of \(0.08 \%\) (a value considered legally drunk in most states). For the cell phone sample, the mean braking reaction time was 779 milliseconds and the standard deviation was 209 milliseconds. For the alcohol sample, the mean breaking reaction time was 849 milliseconds and the standard deviation was \(228 .\) Is there convincing evidence that the mean braking reaction time is different for the population of drivers talking on a cell phone and the population of drivers who have a blood alcohol level of \(0.08 \%\) ? For purposes of this exercise, you can assume that the two samples are representative of the two populations of interest.

Dentists make many people nervous. To see whether such nervousness elevates blood pressure, the blood pressure and pulse rates of 60 subjects were measured in a dental setting and in a medical setting ( \({ }^{4}\) The Effect of the Dental Setting on Blood Pressure Measurement," American Journal of Public Health [1983]: \(1210-1214\) ). For each subject, the difference (dental setting blood pressure minus medical setting blood pressure) was calculated. The (dental - medical) differences were also calculated for pulse rates. Summary statistics follow. $$ \begin{array}{lcc} & & \text { Standard } \\ & \begin{array}{c} \text { Mean } \\ \text { Difference } \end{array} & \begin{array}{c} \text { Deviation of } \\ \text { Differences } \end{array} \\ \text { Systolic Blood Pressure } & 4.47 & 8.77 \\ \text { Pulse (beats/min) } & -1.33 & 8.84 \end{array} $$

The press release titled "Keeping Score When It Counts: Graduation Rates and Academic Progress Rates" (The Institute for Diversity and Ethics in Sport, March 16,2009 ) gave the 2009 graduation rates for African American basketball players and for white basketball players at every NCAA Division I university with a basketball program. Explain why it is not necessary to use a paired- samples \(t\) test to determine if the 2009 mean graduation rate for African American basketball players differs from the 2009 mean graduation rate for white basketball players for NCAA Division I schools.

An individual can take either a scenic route to work or a nonscenic route. She decides that use of the nonscenic route can be justified only if it reduces the mean travel time by more than 10 minutes. a. If \(\mu_{1}\) refers to the mean travel time for scenic route and \(\mu_{2}\) to the mean travel time for nonscenic route, what hypotheses should be tested? b. If \(\mu_{1}\) refers to the mean travel time for nonscenic route and \(\mu_{2}\) to the mean travel time for scenic route, what hypotheses should be tested?
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