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Chapter 13: Problem 20

The uuthors ol the paper "Concordance of Self-Report and Measured Height and Weight of College Students" (Journal of Nutrition, Education and Behavior [2015]: \(94-98\) ) used a pairedsamples \(t\) test to reach the conclusion that male college students tend to over-report both weight and height. This conclusion was based on a sample of 634 male college students selected from eight different universities. The sample mean difference between the reported weight and actual measured weight was 1.2 pounds and the standard deviation of the differences was 5.71 pounds. For purposes of this exercise, you can assume that the sample was representative of male college students. a. Carry out a hypothesis test to determine if there is a significant difference in the mean reported weight and the mean actual weight for male college students. b. For height, the mean difference between the reported height and actual measured height was 0.6 inches and the standard deviation of the differences was 0.8 inches. Carry out a hypothesis test to determine if there is a significant difference in the mean reported height and the mean actual height for male college students. c. Do the conclusions reached in the hypothesis tests of Parts (a) and (b) support the given conclusion that male college students tend to over-report both height and weight? Explain.

Short Answer

Expert verified
In the hypothesis test for weight differences, we rejected the null hypothesis, which implies that there is a significant difference between the mean reported weight and the mean actual weight for male college students. Similarly, in the hypothesis test for height differences, we rejected the null hypothesis, implying that there is a significant difference between the mean reported height and the mean actual height for male college students. Since both tests resulted in rejecting the null hypothesis, these results support the given conclusion that male college students tend to over-report both height and weight.

Step by step solution

01

(Test for weight differences)

: We will be performing a paired-samples t-test to determine if there is a significant difference in the mean reported weight and the mean actual weight for male college students. Step 1: State the null hypothesis and alternative hypothesis. \(H_0\): There is no difference in the mean reported weight and the mean actual weight of male college students; i.e., \(\mu_d = 0\). \(H_1\): There is a difference in the mean reported weight and the mean actual weight of male college students; i.e., \(\mu_d \neq 0\). Step 2: Determine the test statistic. The sample size is \(n = 634\), sample mean difference in weight is \(\bar{d} = 1.2\), and the standard deviation of the differences is \(s_d = 5.71\). The test statistic is computed as follows: \(t = \frac{\bar{d}-\mu_d}{\frac{s_d}{\sqrt{n}}} = \frac{1.2-0}{\frac{5.71}{\sqrt{634}}}\). Step 3: Compute the p-value. Using a t-distribution calculator or t-table, find the p-value corresponding to the computed test statistic with \(n-1\) degrees of freedom. Step 4: Make a decision. Compare the p-value with the significance level \(\alpha\) (typically 0.05). If the p-value is less than \(\alpha\), reject the null hypothesis; otherwise, fail to reject the null hypothesis. Step 5: Interpret the results. Based on your decision in step 4, state whether there is a significant difference in the mean reported weight and the mean actual weight for male college students.
02

(Test for height differences)

: Similarly, we will be performing a paired-samples t-test to determine if there is a significant difference in the mean reported height and the mean actual height for male college students. Step 1: State the null hypothesis and alternative hypothesis. \(H_0\): There is no difference in the mean reported height and the mean actual height of male college students; i.e., \(\mu_d = 0\). \(H_1\): There is a difference in the mean reported height and the mean actual height of male college students; i.e., \(\mu_d \neq 0\). Step 2: Determine the test statistic. The sample size is \(n = 634\), the sample mean difference in height is \(\bar{d} = 0.6\), and the standard deviation of the differences is \(s_d = 0.8\). The test statistic is computed as follows: \(t = \frac{\bar{d}-\mu_d}{\frac{s_d}{\sqrt{n}}} = \frac{0.6-0}{\frac{0.8}{\sqrt{634}}}\). Step 3: Compute the p-value. Using a t-distribution calculator or t-table, find the p-value corresponding to the computed test statistic with \(n-1\) degrees of freedom. Step 4: Make a decision. Compare the p-value with the significance level \(\alpha\) (typically 0.05). If the p-value is less than \(\alpha\), reject the null hypothesis; otherwise, fail to reject the null hypothesis. Step 5: Interpret the results. Based on your decision in step 4, state whether there is a significant difference in the mean reported height and the mean actual height for male college students.
03

(Comparing the conclusions)

: Examine the conclusions made from the hypothesis tests conducted in both parts (a) and (b) and determine whether they provide support for the claim that male college students tend to over-report both height and weight. If both tests resulted in rejecting the null hypothesis, this supports the given conclusion. If either (or both) of the tests did not reject the null hypothesis, the data does not provide sufficient evidence to support the claim that male college students over-report both height and weight.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a structured process used in statistics to determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. It starts by proposing two opposing statements: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\) or \(H_a\)). A test statistic is calculated from the sample data and is used to assess the plausibility of the null hypothesis. The quintessential goal is to analyze if there's enough empirical data to reject the null hypothesis, suggesting that the alternative hypothesis may be accepted.

In our example where researchers are comparing self-reported and measured height and weight of college students, hypothesis testing allows them to evaluate the likelihood that the observed differences are due to chance, as opposed to a systematic overestimation by the students.
Null Hypothesis
The null hypothesis, denoted as \(H_0\), posits that there is no effect or no difference in the general population that the sample data is expected to represent. It is the default position that there is no association between two measured phenomena. In the context of our exercise, the null hypothesis states that the average reported weight (\(\mu_d = 0\)) is the same as the actual weight in the population of male college students. This hypothesis serves as a baseline, which can be contested by the sample data when conducting the paired-samples \(t\)-test.
Alternative Hypothesis
The alternative hypothesis \(H_1\) or \(H_a\) is the statement that there is a statistically significant effect or difference. It contradicts the null hypothesis and is considered to be true if the null hypothesis is rejected. When we formulate \(H_1\) for our paired-samples \(t\)-test, we state that there is a real difference in mean reported and actual weights (\(\mu_d eq 0\)) for male college students. Accepting the alternative hypothesis usually requires evidence from the sample data indicating that the null hypothesis is unlikely.
Statistical Significance
Statistical significance addresses the question of whether any observed differences in data are likely due to chance or to some systematic effect. It is a determination made by calculating the probability that an observed effect would occur if the null hypothesis were true. A statistically significant result is one that is not attributed to chance and thus provides enough evidence to reject the null hypothesis. To determine if the test is significant, the computed \(p\)-value from a statistical test is compared against a predetermined threshold, known as the alpha level (\(\alpha\)), which typically is set at 0.05 or 5%. If the \(p\)-value is less than or equal to \(\alpha\), the result is deemed statistically significant. In our problem, if the \(p\)-value for the paired-samples \(t\)-test regarding the weight and height differences is below 0.05, it would indicate significant evidence against the null hypothesis.
P-Value
The \(p\)-value is a metric used in hypothesis testing to quantify the evidence against the null hypothesis. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one obtained from the sample data, assuming that the null hypothesis is true.
  • If the \(p\)-value is small (typically 鈮� \(\alpha\)), it indicates strong evidence against the null hypothesis, so we reject \(H_0\).
  • If the \(p\)-value is large, it suggests the evidence is not strong enough to reject the null hypothesis.
The exercise requires us to calculate this value using our sample data. Once we find the \(p\)-value for both height and weight differences, we can make an informed decision about whether there is a significant discrepancy between what male college students report and what is actually measured.

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Most popular questions from this chapter

For each of the following hypothesis testing scenarios, indicate whether or not the appropriate hypothesis test would be about a difference in two population means. If not, explain why not.

The humorous puper "Will Humans Swim Faster or Slower in Syrup?" (American Institute of Chemical Engineers Journal [2004]: \(2646-2647\) ) investigated the fluid mechanics of swimming. Twenty swimmers each swam a specified distance in a water-filled pool and in a pool in which the water was thickened with food grade guar gum to create a syruplike consistency. Velocity, in meters per second, was recorded. Values estimated from a graph in the paper are given. The authors of the paper concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level.

Herc's one to sink your teeth into: The authors of the article "Analysis of Food Crushing Sounds During Mastication: Total Sound Level Studies" (Journal of Texture Studies [1990]: \(165-178\) ) studicd the nature of sounds generated during eating. Peak loudness was measured (in decibels at \(20 \mathrm{~cm}\) away \()\) for both open-mouth and closedmouth chewing of potato chips and of tortilla chips. A sample of size 10 was used for each of the four possible combinations (such as closed-mouth potato chip, and so on). We are not making this up! Summary values taken from plots given in the article appear in the accompanying table. For purposes of this exercise, suppose that it is reasonable to regard the peak loudness distributions as approximately normal. a. Construct a 95\% confidence interval for the difference in mean peak loudness between open-mouth and closedmouth chewing of potato chips. Be sure to interpret the resulting interval. b. For closed-mouth chewing (the recommended method!), construct a \(95 \%\) confidence interval for the difference in mean peak loudness between potato chips and tortilla chips.

The National Sleep Foundation surveyed representative samples of adults in six different countries to ask questions about sleeping habits ("2013 International Bedroom Poll Summary of Findings," www.sleepfoundation.org/sites /default/files/RPT495a.pdf, retrieved May 20,2017 ). Each person in a representative sample of 250 adults in each of these countries was asked how much sleep they get on a typical work night. For the United States, the sample mean was 391 minutes, and for Mexico the sample mean was 426 minutes. Suppose that the sample standard deviations were 30 minutes for the U.S. sample and 40 minutes for the Mexico sample. The report concludes that on average, adults in the United States get less sleep on work nights than adults in Mexico. Is this a reasonable conclusion? Support your answer with an appropriate hypothesis test.

Research has shown that, for baseball players, good hip range of motion results in improved performance and decreased body stress. The article "Functional Hip Characteristics of Baseball Pitchers and Position Players" (The American journal of Sports Medicine, \(2010: 383-388\) ) reported on a study of independent samples of 40 professionul pitchers and 40 professional position players. For the pitchers, the sample mean hip range of motion was 75.6 degrees and the sample standard deviation was 5.9 degrees, whercas the sample mean and sample standard deviation for position players were 79.6 degrees and 7.6 degrees, respectively. Assuming that the two samples are representative of professional bascball pitchers and position players, test hypotheses appropriate for determining if there is convincing evidence that the mean range of motion for pitchers is less than the mean for position players.
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