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Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

Chapter 26

Analysis of Variance - all with Video Answers

Educators

Chapter Questions

09:28

Problem 1

A student runs an experiment to test four different popcorn brands, recording the number of kernels left unpopped. She pops measured batches of each brand 3 times, using the same popcorn popper and randomizing the order of the brands. After collecting her data and analyzing the results, she reports that the $F$ -ratio is 13.52 .
a) What are the null and alternative hypotheses?
b) How many degrees of freedom does the treatment sum of squares have? How about the error sum of squares?
c) Assuming that the conditions required for ANOVA are satisfied, what is the P-value? What would you conclude?
d) What else about the data would you like to see in order to check the assumptions and conditions?

Khalida Dawar
Khalida Dawar
Numerade Educator
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Problem 2

A figure skater tried various approaches to her Salchow jump in a designed experiment using 5 different places for her focus (arms, free leg, midsection, takeoff leg, and free). She tried each jump 6 times in random order, using two of her skating partners to judge the jumps on a scale from 0 to 6 . After collecting the data and analyzing the results, she reports that the $F$ -ratio is 7.43 .
a) What are the null and alternative hypotheses?
b) How many degrees of freedom does the treatment sum of squares have? How about the error sum of squares?
c) Assuming that the conditions are satisfied, what is the P-value? What would you conclude?
d) What else about the data would you like to see in order to check the assumptions and conditions?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
09:28

Problem 3

A student runs an experiment to study the effect of three different mufflers on gas mileage. He devises a system so that his Jeep Wagoneer uses gasoline from a one-liter container. He tests each muffler 10 times, carefully recording the number of miles he can go in his Jeep Wagoneer on one liter of gas. After analyzing his data, he reports that the $F$ -ratio is 2.37 with a $P$ -value of 0.0867 .
a) What are the null and alternative hypotheses?
b) How many degrees of freedom does the treatment sum of squares have? How about the error sum of squares?
c) What would you conclude?
d) What else about the data would you like to see in order to check the assumptions and conditions?
e) If your conclusion in part $c$ is wrong, what type of error have you made?

Khalida Dawar
Khalida Dawar
Numerade Educator
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Problem 4

A student interested in improving her dart-throwing technique designs an experiment to test 4 different stances to see whether they affect her accuracy. After warming up for several minutes, she randomizes the order of the 4 stances, throws a dart at a target using each stance, and measures the distance of the dart in centimeters from the center of the bull's-eye. She replicates this procedure 10 times. After analyzing the data she reports that the $F$ -ratio is 1.41 .
a) What are the null and alternative hypotheses?
b) How many degrees of freedom does the treatment sum of squares have? How about the error sum of squares?
c) What would you conclude?
d) What else about the data would you like to see in order to check the assumptions and conditions?
e) If your conclusion in part $\mathrm{c}$ is wrong, what type of error have you made?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:53

Problem 5

To shorten the time it takes him to make his favorite pizza, a student designed an experiment to test the effect of sugar and milk on the activation times for baking yeast. Specifically, he tested four different recipes and measured how many seconds it took for the same amount of dough to rise to the top of a bowl. He randomized the order of the recipes and replicated each treatment 4 times.

Here are the boxplots of activation times from the four recipes:
a) State the hypotheses about the recipes (both numerically and in words).
b) Assuming that the assumptions for inference are satisfied, perform the hypothesis test and state your conclusion. Be sure to state it in terms of activation times and recipes.
c) Would it be appropriate to follow up this study with multiple comparisons to see which recipes differ in their mean activation times? Explain.

James Kiss
James Kiss
Numerade Educator
02:23

Problem 6

A student performed an experiment with three different grips to see what effect it might have on the distance of a backhanded Frisbee throw. She tried it with her normal grip, with one finger out, and with the Frisbee inverted. She measured in paces how far her throw went. The boxplots and the ANOVA table for the three grips are shown below:
a) State the hypotheses about the grips.
b) Assuming that the assumptions for inference are satisfied, perform the hypothesis test and state your conclusion. Be sure to state it in terms of Frisbee grips and distance thrown.
c) Would it be appropriate to follow up this study with multiple comparisons to see which grips differ in their mean distance thrown? Explain.

James Kiss
James Kiss
Numerade Educator
01:40

Problem 7

Here are boxplots, first seen in Chapter 4 Exercise $28,$ that show the relationship between the number of cylinders a car's engine has and the car's fuel economy.
a) State the null and alternative hypotheses that you might consider for these data.
b) Do the conditions for an ANOVA seem to be met here? Why or why not?

James Kiss
James Kiss
Numerade Educator
03:15

Problem 8

Here are bottle prices (in dollars) of wines produced by wineries along three of the Finger Lakes.
a) What null and alternative hypotheses would you test for these data? Talk about prices and location, not symbols.
b) Do the conditions for an ANOVA seem to be met here? Why or why not?

James Kiss
James Kiss
Numerade Educator
02:14

Problem 9

A bank is studying the time that it takes 6 of its tellers to serve an average customer. Customers line up in the queue and then go to the next available teller. Here is a boxplot of the last 200 customers and the times it took each teller:
a) What are the null and alternative hypotheses?
b) What do you conclude?
c) Would it be appropriate to run a multiple comparisons test (for example, a Bonferroni test) to see which tellers differ from each other? Explain.

James Kiss
James Kiss
Numerade Educator
02:48

Problem 10

A researcher investigated four different word lists for use in hearing assessment. She wanted to know whether the lists were equally difficult to understand in the presence of a noisy background. To find out, she tested 96 subjects with normal hearing randomly assigning 24 to each of the four word lists and measured the number of words perceived correctly in the presence of background noise. Here are the boxplots of the four lists:
a) What are the null and alternative hypotheses?
b) What do you conclude?
c) Would it be appropriate to run a multiple comparisons test (for example, a Bonferroni test) to see which lists differ from each other in terms of mean percent correct? Explain.

James Kiss
James Kiss
Numerade Educator
01:55

Problem 11

In Chapter $4,$ Exercise $32,$ we saw a survey of 1021 school-age children conducted by randomly selecting children from several large urban elementary schools. Two of the questions concerned eye and hair color. In the survey, the following codes were used:
$$
\begin{array}{l|l}
\hline \text { Hair Color } & \text { Eye Color } \\
\hline 1=\text { Blond } & 1=\text { Blue } \\
2=\text { Brown } & 2=\text { Green } \\
3=\text { Black } & 3=\text { Brown } \\
4=\operatorname{Red} & 4=\text { Grey } \\
5=0 \text { ther } & 5=0 \text { ther }
\end{array}
$$
The students analyzing the data were asked to study the relationship between eye and hair color. They produced this plot:
They then ran an Analysis of Variance with Eye Color as the response and Hair Color as the factor. The ANOVA table they produced follows:
What suggestions do you have for the Statistics students? What alternative analysis might you suggest?

James Kiss
James Kiss
Numerade Educator
01:34

Problem 12

The intern from the marketing department at the Holes $\mathrm{R}$ Us online piercing salon (Chapter 3 , Exercise 55 ) has recently finished a study of the company's 500 customers. He wanted to know whether people's ZIP codes vary by the last product they bought.
They have 16 different products, and the ANOVA table of ZIP code by product showed the following:
(Nine customers were not included because of missing ZIP code or product information.)

What criticisms of the analysis might you make? What alternative analysis might you suggest?

James Kiss
James Kiss
Numerade Educator
02:34

Problem 13

Yogurt An experiment to determine the effect of several methods of preparing cultures for use in commercial yogurt was conducted by a food science research group. Three batches of yogurt were prepared using each of three methods: traditional, ultrafiltration, and reverse osmosis. A trained expert then tasted each of the 9 samples, presented in random order, and judged them on a scale from 1 to 10 . A partially completed Analysis of Variance table of the data follows.

James Kiss
James Kiss
Numerade Educator
02:17

Problem 14

Particulate matter is a serious form of air pollution often arising from industrial production. One way to reduce the pollution is to put a filter, or scrubber, at the end of the smokestack to trap the particulates. An experiment to determine which smokestack scrubber design is best was run by placing four scrubbers of different designs on an industrial stack in random order. Each scrubber was tested 5 times. For each run, the same material was produced, and the particulate emissions coming out of the scrubber were measured (in parts per billion). A partially completed Analysis of Variance table of the data follows on the next page.
a) Calculate the mean square of the treatments and the mean square of the error.
b) Form the $F$ -statistic by dividing the two mean squares.
c) The P-value of this $F$ -statistic turns out to be 0.0001204 . What does this say about the null hypothesis of equal means?
d) What assumptions have you made in order to answer part c?
e) What would you like to see in order to justify the conclusions of the $F$ -test?
f) What is the average size of the error standard deviation in particulate emissions?

James Kiss
James Kiss
Numerade Educator
02:26

Problem 15

A student wants to investigate the effects of real vs. substitute eggs on his favorite brownie recipe. He enlists the help of 10 friends and asks them to rank each of 8 batches on a scale from 1 to 10 . Four of the batches were made with real eggs, four with substitute eggs. The judges tasted the brownies in random order. Here is a boxplot of the data:
The mean score for the real eggs was 6.78 with a standard deviation of 0.651 . The mean score for the substitute eggs was 4.66 with a standard deviation of 0.395 .
a) What are the null and alternative hypotheses?
b) What do you conclude from the ANOVA table?
c) Do the assumptions for the test seem to be reasonable?
d) Perform a two-sample pooled $t$ -test of the difference. What $P$ -value do you get? Show that the square of the $t$ -statistic is the same (to rounding error) as the $F$ -ratio.

James Kiss
James Kiss
Numerade Educator
02:12

Problem 16

In a statement to a Senate Public Works Committee, a senior executive of Texaco, Inc., cited a study on the effectiveness of auto filters on reducing noise. Because of concerns about performance, two types of filters were studied, a standard silencer and a new device developed by the Associated Octel Company. Here are the boxplots from the data on noise reduction (in decibels) of the two filters. Type $1=$ standard; Type $2=$ Octel.

a) What are the null and alternative hypotheses?
b) What do you conclude from the ANOVA table?
c) Do the assumptions for the test seem to be reasonable?
d) Perform a two-sample pooled $t$ -test of the difference. What P-value do you get? Show that the square of the $t$ -statistic is the same (to rounding error) as the $F$ -ratio.

James Kiss
James Kiss
Numerade Educator
01:54

Problem 17

A school district superintendent wants to test a new method of teaching arithmetic in the fourth grade at his 15 schools. He plans to select 8 students from each school to take part in the experiment, but to make sure they are roughly of the same ability, he first gives a test to all 120 students. Here are the scores of the test by school:
a) What are the null and alternative hypotheses?
b) What does the ANOVA table say about the null hypothesis? (Be sure to report this in terms of scores and schools.)
c) An intern reports that he has done $t$ -tests of every school against every other school and finds that several of the schools seem to differ in mean score. Does this match your finding in part b? Give an explanation for the difference, if any, of the two results.

James Kiss
James Kiss
Numerade Educator
01:40

Problem 18

Fertilizers A biology student is studying the effect of 10 different fertilizers on the growth of mung bean sprouts. She sprouts 12 beans in each of 10 different petri dishes, and adds the same amount of fertilizer to each dish. After one week she measures the heights of the 120 sprouts in millimeters. Here are boxplots and an ANOVA table of the data:

a) What are the null and alternative hypotheses?
b) What does the ANOVA table say about the null hypothesis? (Be sure to report this in terms of heights and fertilizers).
c) Her lab partner looks at the same data and says that he did $t$ -tests of every fertilizer against every other fertilizer and finds that several of the fertilizers seem to have significantly higher mean heights. Does this match your finding in part b? Give an explanation for the difference, if any, between the two results.

James Kiss
James Kiss
Numerade Educator
02:21

Problem 19

We first saw data on breakfast cereals in Chapter 7 . Supermarkets often place similar types of cereal on the same supermarket shelf. We have data on the shelf as well as the sugar, sodium, and calorie content of 77 cereals. Does sugar content vary by shelf? At the top of the next column is a boxplot and an ANOVA table for the 77 cereals.
a) What are the null and alternative hypotheses?
b) What does the ANOVA table say about the null hypothesis? (Be sure to report this in terms of Sugars and Shelves.)
c) Can we conclude that cereals on shelf 2 have a higher mean sugar content than cereals on shelf 3 ? Can we conclude that cereals on shelf 2 have a higher mean sugar content than cereals on shelf $1 ?$ What $\mathrm{can}$ we conclude?
d) To check for significant differences between the shelf means, we can use a Bonferroni test, whose results are shown below. For each pair of shelves, the difference is shown along with its standard error and significance level. What does it say about the questions in part c?

James Kiss
James Kiss
Numerade Educator
01:32

Problem 20

We also have data on the protein content of the cereals in Exercise 19 by their shelf number. Here are the boxplot and ANOVA table:
a) What are the null and alternative hypotheses?
b) What does the ANOVA table say about the null hypothesis? (Be sure to report this in terms of protein content and shelves.)
c) Can we conclude that cereals on shelf 2 have a lower mean protein content than cereals on shelf 3 ? Can we conclude that cereals on shelf 2 have a lower mean protein content than cereals on shelf 1 ? What $\mathrm{can}$ we conclude?
d) To check for significant differences between the shelf means we can use a Bonferroni test, whose results are shown below. For each pair of shelves, the difference is shown along with its standard error and significance level. What does it say about the questions in part $\mathrm{c}$ ?

James Kiss
James Kiss
Numerade Educator
03:02

Problem 21

To see how much of a difference time of day made on the speed at which he could download files, a college sophomore performed an experiment. He placed a file on a remote server and then proceeded to download it at three different time periods of the day. He downloaded the file 48 times in all, 16 times at each Time of Day, and recorded the Time in seconds that the download took.
a) State the null and alternative hypotheses, being careful to talk about download Time and Time of Day as well as parameters.
b) Perform an ANOVA on these data. What can you conclude?
c) Check the assumptions and conditions for an ANOVA. Do you have any concerns about the experimental design or the analysis?
d) (Optional) Perform a multiple comparisons test to determine which times of day differ in terms of mean download time.

James Kiss
James Kiss
Numerade Educator
02:59

Problem 22

A pharmaceutical company tested three formulations of a pain relief medicine for migraine headache sufferers. For the experiment, 27 volunteers were selected and 9 were randomly assigned to one of three drug formulations. The subjects were instructed to take the drug during their next migraine headache episode and to report their pain on a scale of $1=$ no pain to $10=$ extreme pain 30 minutes after taking the drug.
a) State the null and alternative hypotheses, being careful to talk about Drug and Pain levels as well as parameters.
b) Perform an ANOVA on these data. What can you conclude?
c) Check the assumptions and conditions for an ANOVA. Do you have any concerns about the experimental design or the analysis?
d) (Optional) Perform a multiple comparisons test to determine which drugs differ in terms of mean pain level reported.

James Kiss
James Kiss
Numerade Educator