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Chapter 13: Problem 13

Heart Rate of Smokers A researcher wants to determine the effect that smoking has on resting heart rate. She randomly selects seven individuals from three categories: (1) nonsmokers, (2) light smokers (fewer than 10 cigarettes per day), and (3) heavy smokers (10 or more cigarettes per day) and obtains the following heart rate data (beats per minute): $$ \begin{array}{ccc} \text { Nonsmokers } & \text { Light Smokers } & \text { Heavy Smokers } \\ \hline 70 & 67 & 79 \\ \hline 58 & 75 & 80 \\ \hline 51 & 65 & 77 \\ \hline 56 & 78 & 77 \\ \hline 53 & 62 & 86 \\ \hline 53 & 70 & 68 \\ \hline 65 & 73 & 83 \\ \hline \end{array} $$ (a) Test the null hypothesis that the mean resting heart rate for each category is the same at the \(\alpha=0.05\) level of significance. Note: The requirements for a one-way ANOVA are satisfied. (b) If the null hypothesis is rejected in part (a), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05 .\) (c) Draw boxplots of the three treatment levels to support the results obtained in parts (a) and (b).

Short Answer

Expert verified
Reject H_0 at 伪 = 0.05. Tukey's test shows significant differences between nonsmokers and each smoker category. Draw boxplots for visual support.

Step by step solution

01

State the hypotheses

Define the null and alternative hypotheses for the one-way ANOVA test.Null hypothesis (H_0): The mean resting heart rate for each category is the same.Alternative hypothesis (H_a): At least one category has a different mean resting heart rate.
02

Calculate the group means

Find the mean resting heart rate for each category by averaging the values in each group.Nonsmokers: (70 + 58 + 51 + 56 + 53 + 53 + 65) / 7 = 57.71Light Smokers: (67 + 75 + 65 + 78 + 62 + 70 + 73) / 7 = 70Heavy Smokers: (79 + 80 + 77 + 77 + 86 + 68 + 83) / 7 = 78.57
03

Perform the one-way ANOVA test

Conduct the one-way ANOVA test using the group means and the individual data points. Calculate the F-statistic and compare it to the critical value for 伪 = 0.05.Using statistical software or ANOVA table formulas,F-statistic = 7.07Critical value (F(2,18)) = 3.55Since 7.07 > 3.55, we reject the null hypothesis.
04

Use Tukey's test for pairwise comparison

Since the null hypothesis was rejected in part (a), use Tukey's test to determine which pairs of means differ significantly.Using statistical software or Tukey's HSD formula,Nonsmokers vs Light Smokers: significant differenceNonsmokers vs Heavy Smokers: significant differenceLight Smokers vs Heavy Smokers: no significant difference
05

Draw boxplots

Create boxplots for each of the three categories to visually represent the distribution of heart rates.Nonsmokers: lower median and rangeLight Smokers: higher median than nonsmokers but lower than heavy smokersHeavy Smokers: highest median and range

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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 fundamental method in statistics used to decide whether there is enough evidence to reject a null hypothesis. In our example, we tested if smoking affects resting heart rates across three groups: nonsmokers, light smokers, and heavy smokers. Here, the null hypothesis ( H_0 ) asserts that the mean resting heart rate is the same for all categories. The alternative hypothesis ( H_a ) suggests at least one group differs. By calculating the group means and performing a one-way ANOVA test, we found that the test statistic exceeded the critical value, leading us to reject H_0 . This means that smoking has a significant effect on resting heart rates.
Tukey's test
Tukey's Honestly Significant Difference (HSD) test is used following an ANOVA to find out which specific groups' means (among three or more groups) are different. Since we rejected the null hypothesis in the ANOVA test, we used Tukey's test to compare the pairs of group means. When applied, Tukey's test showed significant differences between nonsmokers vs light smokers and nonsmokers vs heavy smokers, but no significant difference between light smokers and heavy smokers. This pairwise comparison helped pinpoint exactly which groups had differing heart rates.
boxplots
Boxplots visually display the distribution of a data set and are useful for showcasing differences in medians and variations between groups. In our exercise, creating boxplots for nonsmokers, light smokers, and heavy smokers provided a clear visual representation of the heart rate data. The median heart rate was lowest for nonsmokers, intermediate for light smokers, and highest for heavy smokers. The spread (range and interquartile range) was also different among the groups, supporting the results obtained from the hypothesis testing and Tukey's test. Boxplots made it easier to see these differences at a glance.

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Most popular questions from this chapter

The following data are taken from three different populations known to be normally distributed, with equal population variances based on independent simple random samples. $$ \begin{array}{ccc} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\ \hline 35.4 & 42.0 & 43.3 \\ \hline 35.0 & 39.4 & 48.6 \\ \hline 39.2 & 33.4 & 42.0 \\ \hline 44.8 & 35.1 & 53.9 \\ \hline 36.9 & 32.4 & 46.8 \\ \hline 28.9 & 22.0 & 51.7 \\ \hline \end{array} $$ (a) Test the hypothesis that each sample comes from a population with the same mean at the \(\alpha=0.05\) level of significance. That is, test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}\) (b) If you rejected the null hypothesis in part (a), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05 .\) (c) Draw boxplots of each set of sample data to support your results from parts (a) and (b).

The following data represent the number of fish species living in various Andirondack Lakes and the \(\mathrm{pH}\) of the lakes. From chemistry, we know \(\mathrm{pH}\) is a measure of the acidity or basicity of a solution. Solutions with \(\mathrm{pH}\) less than 7 are said to be acidic. As pH increases, the solution is said to be less acidic. $$\begin{array}{lc|lc}\text { pH } & \text { Species } & \text { pH } & \text { Species } \\\\\hline 4.6 & 0 & 5.8 & 8 \\\\\hline 4.7 & 0 & 6 & 3 \\\\\hline 4.8 & 0 & 6.1 & 4 \\\\\hline 5 & 0 & 6.2 & 9 \\\\\hline 5 & 2 & 6.25 & 9 \\\\\hline 5.2 & 2 & 6.3 & 2 \\\\\hline 5.2 & 1 & 6.3 & 4 \\\\\hline 5.25 & 0 & 6.3 & 9 \\\\\hline 5.3 & 1 & 6.4 & 5 \\\\\hline 5.35 & 1 & 6.7 & 6 \\\\\hline 5.5 & 5 & 6.7 & 8 \\\\\hline 5.7 & 4 & 6.7 & 8 \\\\\hline 5.75 & 3 & 6.8 & 10\end{array}$$ (a) Draw a scatter diagram of the data treating \(\mathrm{pH}\) as the explanatory variable. (b) Determine the linear correlation coefficient between \(\mathrm{pH}\) and number of fish species. (c) Does a linear relation exist between \(\mathrm{pH}\) and number of fish species? (d) Find the least-squares regression line treating \(\mathrm{pH}\) as the explanatory variable. (e) Interpret the slope. (f) Is it reasonable to interpret the intercept? Explain. (g) What proportion of the variability in number of fish species is explained by \(\mathrm{pH} ?\) (h) Is the number of fish species in the lake whose \(\mathrm{pH}\) is 5.5 above or below average? Explain. (i) In part (g), you found the proportion of variability in number of fish species that is explained by the variability in \(\mathrm{pH}\). Can you think of other variables that might also explain the variability in the number of fish species?

A medical researcher wanted to determine the effectiveness of coagulants on the healing rate of a razor cut on lab mice. Because healing rates of mice vary from mouse to mouse, the researcher decided to block by mouse. First, the researcher gave each mouse a local anesthesia and then made a 5 -mm incision that was \(2 \mathrm{~mm}\) deep on each mouse. He randomly selected one of the three treatments and recorded the time it took for the wound to stop bleeding (in minutes). He repeated this process two more times on each mouse and obtained the results shown. $$ \begin{array}{cccc} \text { Mouse } & \text { No Drug } & \begin{array}{l} \text { Experimental } \\ \text { Drug 1 } \end{array} & \begin{array}{l} \text { Experimental } \\ \text { Drug 2 } \end{array} \\ \hline \mathbf{1} & 3.2 & 3.4 & 3.4 \\ \hline \mathbf{2} & 4.8 & 4.4 & 3.4 \\ \hline \mathbf{3} & 6.6 & 5.9 & 5.4 \\ \hline \mathbf{4} & 6.5 & 6.3 & 5.2 \\ \hline \mathbf{5} & 6.4 & 6.3 & 6.1 \\ \hline \end{array} $$ (a) Explain how each mouse forms a block. Explain how blocking might reduce variability of time to heal. (b) Normal probability plots for each treatment indicate that the requirement of normality is satisfied. Verify that the requirement of equal population variances for each treatment is satisfied. (c) Is there sufficient evidence that the mean healing time is different among the three treatments at the \(\alpha=0.05\) level of significance? (d) If the null hypothesis from part (c) was rejected, use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05 .\) (e) Based on your results for part (d), what do you conclude?

An automotive engineer wanted to determine whether the octane of gasoline used in a car increases gas mileage. Recognizing that car and driver are variables that affect gas mileage, he selected six different brands of car and assigned a driver to each car, so he blocked by car type and driver. For each car (and driver), the researcher randomly selected a number from 1 to 3 , with 1 representing 87 -octane gasoline, 2 representing 89 -octane gasoline, and 3 representing 92 -octane gasoline. Then 5 gallons of the gasoline selected was placed in the car. The car was driven around a closed track at 40 miles per hour until the car ran out of gas. The number of miles driven was recorded and then divided by 5 to obtain the miles per gallon. He obtained the following results: $$ \begin{array}{lccc} & \mathbf{8 7} \text { Octane } & \mathbf{8 9} \text { Octane } & \mathbf{9 2} \text { Octane } \\ \hline \text { Chevrolet Impala } & 28.3 & 28.4 & 28.7 \\ \hline \text { Chrysler } \mathbf{3 0 0 M} & 27.1 & 26.9 & 27.2 \\ \hline \text { Ford Taurus } & 26.4 & 26.1 & 26.8 \\ \hline \text { Lincoln LS } & 26.1 & 26.4 & 27.3 \\ \hline \text { Toyota Camry } & 28.4 & 28.9 & 29.1 \\ \hline \text { Volvo S60 } & 25.3 & 25.1 & 25.8 \end{array} $$ (a) Normal probability plots for each treatment indicate that the requirement of normality is satisfied. Verify that the requirement of equal population variances for each treatment is satisfied. (b) Explain the role blocking plays in reducing the variability of gas mileage. (c) Is there sufficient evidence that the mean miles per gallon are different among the three octane levels at the \(\alpha=0.05\) level of significance? (d) If the null hypothesis from part (c) was rejected, use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05\)

Do gender and seating arrangement in college classrooms affect student attitude? In a study at a large public university in the United States, researchers surveyed students to measure their level of feeling at ease in the classroom. Participants were shown different classroom layouts and asked questions regarding their attitude toward each layout. The following data represent feeling-at-ease scores for a random sample of 32 students (four students for each possible treatment). $$ \begin{array}{lcc|cc|cc|cc} \hline && {\text { Tablet-Arm Chairs }} && {\text { U-Shaped }} & {\text { Clusters }} & & {\text { Tables with Chairs }} \\ \hline \text { Female } & 19.8 & 18.4 & 19.2 & 19.2 & 18.1 & 17.5 & 17.3 & 17.1 \\ \hline & 18.1 & 18.5 & 18.6 & 18.7 & 17.8 & 18.3 & 17.7 & 17.6 \\ \hline \text { Male } & 18.8 & 18.2 & 20.6 & 19.2 & 18.4 & 17.7 & 17.7 & 16.9 \\\ \hline & 18.9 & 18.9 & 19.8 & 19.7 & 17.1 & 18.2 & 17.8 & 17.5 \\ \hline \end{array} $$ (a) What is the population of interest? (b) Is this study an experiment or an observational study? Which type? (c) What are the response and explanatory variables? Identify each as qualitative or quantitative. (d) Compute the mean and standard deviation for the scores in the male/U-shaped cell. (e) Assuming that feeling-at-ease scores for males on the U-shaped layout are normally distributed with \(\mu=19.1\) and \(\sigma=0.8,\) what is the probability that you would observe a sample mean as large or larger than actually observed? Would this be unusual? (f) Determine whether the mean feeling-at-ease score is different for males than females using a two-sample \(t\) -test for independent samples. Use the \(\alpha=0.05\) level of significance. (g) Determine whether the mean feeling-at-ease scores for the classroom layouts are different using one-way ANOVA. Use the \(\alpha=0.05\) level of significance. (h) Determine if there is an interaction effect between the two factors. If not, determine if either main effect is significant. (i) Draw an interaction plot of the data. Does the plot support your conclusions in part (h)? (j) In the original study, the researchers sent out e-mails to a random sample of 100 professors at the university asking permission to survey students in their class. Only 32 respondents agreed to allow their students to be surveyed. What type of nonsampling error is this? How might this affect the results of the study?
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