Question

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 \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\ \text { Light } & 2 & 116.18 & 58.09 & 8.96 & 0.001 \\ \text { Error } & 25 & 162.10 & 6.48 & & \\ \text { Total } & 27 & 278.28 & & & \end{array}\)

Short answer

There is a statistically significant difference in weight gain between at least one pair of the light conditions (DM, LD, and LL) as the p-value (0.001) is less than 0.05. To identify the specific pairs that are significantly different, post-hoc comparison is required.

Step-by-step solution

Interpreting the ANOVA table

The ANOVA table is read as follows: For the Light variable (referring to DM, LD, and LL), the degrees of freedom (DF) are 2 (The count of treatments minus 1). The sum of squares (SS), a measure of total variability, is 116.18. The mean square (MS), a measure of average variability, is 58.09. The F-value is 8.96 and the p-value is 0.001.

Determining significance

The aim is to test if there are differences between the means of the three groups. An F-value is calculated and the corresponding p-value is observed to find out whether the differences are due to random chance or are statistically significant. In this case, the p-value is 0.001 which is less than 0.05. Therefore, there is a significant difference in the three light conditions.

Identifying the significantly different groups

Although the groups are significantly different, the ANOVA test doesn't show which specific groups were significantly different from each other. To find the details, a post-hoc test is conducted. Since this post-hoc test results are not provided, it can only be concluded that at least one pair of groups are significantly different, but without additional information, the specific groups can't be identified.