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Chapter 8: Problem 28

The mice in the study had body mass measured throughout the study. Computer output showing body mass gain (in grams) after 4 weeks for each of the three light conditions is shown, and a dotplot of the data is given in Figure 8.6 . \(\begin{array}{lrrr}\text { Level } & \mathrm{N} & \text { Mean } & \text { StDev } \\ \text { DM } & 10 & 7.859 & 3.009 \\ \text { LD } & 8 & 5.926 & 1.899 \\ \text { LL } & 9 & 11.010 & 2.624\end{array}\) (a) In the sample, which group of mice gained the most, on average, over the four weeks? Which gained the least? (b) Do the data appear to meet the requirement of having standard deviations that are not dramatically different? (c) The sample sizes are small, so we check that the data are relatively normally distributed. We see in Figure 8.6 that we have no concerns about the DM and LD samples. However, there is an outlier for the LL sample, at 17.4 grams. We proceed as long as the \(z\) -score for this value is within 卤3 . Find the \(z\) -score. Is it appropriate to proceed with ANOVA? (d) What are the cases in this analysis? What are the relevant variables? Are the variables categorical or quantitative?

Short Answer

Expert verified
a) The LL group of mice gained the most weight on average and the LD group gained the least weight on average. b) The standard deviations are not dramatically different. c) Assuming we have calculated the Z-score for the outlier in the LL sample, if it is within 卤3, we proceed with ANOVA. d) The cases are the different mice while the light conditions and body mass changes are the variables. Light conditions is a categorical variable and body mass change is a quantitative variable.

Step by step solution

01

Identify Body Mass Variation

We can identify which group of mice gained the most and the least mass by looking at the 'Mean' column in the table. The group with the highest mean gained the most weight, while the group with the lowest mean gained the least weight.
02

Check Standard Deviations

We need to see if the standard deviations are not dramatically different. We do this by checking the 'StDev' column in the table. If the standard deviations of each group are relatively close to each other, we can say that the requirement is met.
03

Calculating the Z-Score

The Z-score of a particular data point refers to how many standard deviations it's away from the mean. We need to calculate the Z-score for the outlier of the LL sample at 17.4 grams. The formula to calculate the Z-score is: Z = (X - 渭) / 蟽, where X is the data point, 渭 is the mean, and 蟽 is the standard deviation. If the Z-score is within 卤3, we can proceed with ANOVA.
04

Identifying Cases and Variables

The cases in this analysis are the different mice, while the relevant variables are the light conditions (DM, LD, LL) and the body mass changes. Light conditions (DM, LD, LL) is a categorical variable as it places mice into distinct groups. Body mass change is a quantitative variable as it can be measured and exists on a numerical scale.

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Key Concepts

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

Understanding Body Mass Variation
Body mass variation refers to the change in body mass observed within a group over a specified period. In our study of mice, we look at how their body mass changed over four weeks under different light conditions. To identify which group gained the most or least weight, we examine the mean values for each group in the provided table. The mean represents the average gain in body mass for the group. In the given example:
  • DM group has a mean of 7.859 grams.
  • LD group has a mean of 5.926 grams.
  • LL group has a mean of 11.010 grams.
Hence, the LL group gained the most on average, while the LD group gained the least. Understanding the variation in body mass helps researchers discern how environmental factors, such as light conditions, affect biological changes.
Standard Deviation Analysis
Standard deviation is a measure of the amount of variation or dispersion of a set of values. In other words, it indicates how much individual data points deviate from the mean. For our analysis, we check whether the standard deviations of the groups are similar, which is crucial for reliable ANOVA results. Looking at the table:
  • DM group has a standard deviation of 3.009.
  • LD group has a standard deviation of 1.899.
  • LL group has a standard deviation of 2.624.
These values suggest not much dramatic difference among the groups, supporting that the assumption for ANOVA is met. If one group had a considerably larger standard deviation, it could imply a higher variation, possibly affecting the analysis. Thus, evaluating standard deviations is critical before proceeding with statistical tests.
Categorical vs Quantitative Variables
In statistical analysis, it is essential to distinguish between categorical and quantitative variables. Categorical variables categorize data into groups, while quantitative variables measure characteristics numerically. In our mouse study:
  • The light conditions (DM, LD, LL) are categorical because they classify the mice into distinct environmental settings. Each setting can be seen as a category in the experiment.
  • The body mass gain is quantitative since it is a measurable entity represented by numerical values.
Understanding these differences helps in selecting the right statistical methods for analysis. Categorical variables often dictate the grouping in tests like ANOVA, while quantitative data provide the measurable outcomes being compared.

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Most popular questions from this chapter

Exercises 8.46 to 8.52 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } & \\ \text { A } & 6 & 86.833 & 5.231 & \\ \text { B } & 6 & 76.167 & 6.555 & \\ \text { C } & 6 & 80.000 & 9.230 & \\ \text { D } & 6 & 69.333 & 6.154 & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 3 & 962.8 & 320.9 & 6.64 & 0.003 \\ \text { Error } & 20 & 967.0 & 48.3 & & \\ \text { Total } & 23 & 1929.8 & & & \end{array}\) Is there evidence for a difference in the population means of the four groups? Justify your answer using specific value(s) from the output.

Data 4.1 introduces a study in mice showing that even low-level light at night can interfere with normal eating and sleeping cycles. In the full study, mice were randomly assigned to live in one of three light conditions: LD had a standard light/dark cycle, LL had bright light all the time, and DM had dim light when there normally would have been darkness. Exercises 8.28 to 8.34 in Section 8.1 show that the groups had significantly different weight gain and time of calorie consumption. In Exercises 8.55 and \(8.56,\) we revisit these significant differences. Body Mass Gain Computer output showing body mass gain (in grams) for the mice after four weeks in each of the three light conditions is shown, along with the relevant ANOVA output. Which light conditions give significantly different mean body mass gain? \(\begin{array}{lrrr}\text { Level } & \mathrm{N} & \text { Mean } & \text { StDev } \\ \text { DM } & 10 & 7.859 & 3.009 \\ \text { LD } & 9 & 5.987 & 1.786 \\ \text { LL } & 9 & 11.010 & 2.624\end{array}\) One-way ANOVA: BM4Gain versus Light \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\ \text { Light } & 2 & 116.18 & 58.09 & 8.96 & 0.001 \\ \text { Error } & 25 & 162.10 & 6.48 & & \\ \text { Total } & 27 & 278.28 & & & \end{array}\)

Exercises 8.41 to 8.45 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } \\ \text { A } & 5 & 10.200 & 2.864 \\ \text { B } & 5 & 16.800 & 2.168 \\ \text { C } & 5 & 10.800 & 2.387\end{array}\) \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 2 & 133.20 & 66.60 & 10.74 & 0.002 \\ \text { Error } & 12 & 74.40 & 6.20 & & \\ \text { Total } & 14 & 207.60 & & & \end{array}\) Is there sufficient evidence of a difference in the population means of the three groups? Justify your answer using specific value(s) from the output.

Researchers hypothesized that the increased weight gain seen in mice with light at night might be caused by when the mice are eating. (As we have seen in the previous exercises, it is not caused by changes in amount of food consumed or activity level.) Perhaps mice with light at night eat a greater percentage of their food during the day, when they normally should be sleeping. Conditions for ANOVA are met and computer output for the percentage of food consumed during the day for each of the three light conditions is shown.\(\begin{array}{lrrr}\text { Level } & \mathrm{N} & \text { Mean } & \text { StDev } \\ \text { DM } & 10 & 55.516 & 10.881 \\ \text { LD } & 8 & 36.028 & 8.403 \\ \text { LL } & 9 & 76.573 & 9.646\end{array}\) (a) For mice in this sample on a standard light/dark cycle, what is the average percent of food consumed during the day? What percent is consumed at night? What about mice that had dim light at night? (b) Is there evidence that light at night influences when food is consumed by mice? Justify your answer with a p-value. Can we conclude that there is a cause-and-effect relationship?

Color affects us in many ways. For example, Exercise C.92 on page 498 describes an experiment showing that the color red appears to enhance men's attraction to women. Previous studies have also shown that athletes competing against an opponent wearing red perform worse, and students exposed to red before a test perform worse. \(^{3}\) Another study \(^{4}\) states that "red is hypothesized to impair performance on achievement tasks, because red is associated with the danger of failure." In the study, US college students were asked to solve 15 moderately difficult, five-letter, single-solution anagrams during a 5-minute period. Information about the study was given to participants in either red, green, or black ink just before they were given the anagrams. Participants were randomly assigned to a color group and did not know the purpose of the experiment, and all those coming in contact with the participants were blind to color group. The red group contained 19 participants and they correctly solved an average of 4.4 anagrams. The 27 participants in the green group correctly solved an average of 5.7 anagrams and the 25 participants in the black group correctly solved an average of 5.9 anagrams. Work through the details below to test if performance is different based on prior exposure to different colors. (a) State the hypotheses. (b) Use the fact that sum of squares for color groups is 27.7 and the total sum of squares is 84.7 to complete an ANOVA table and find the F-statistic. (c) Use the F-distribution to find the p-value. (d) Clearly state the conclusion of the test.
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