Question

In Data 4.1 on page 258, we examine a study in which mice were randomly assigned to either a normal light/dark cycle or to have a light on around the clock. According to the paper describing that study, "The global increase in the prevalence of obesity and metabolic disorders coincides with the increase of exposure to light at night." The study is examining whether light at night plays a causal role in the obesity epidemic. In Data 4.1, we examine the effectof hte different light conditions on body mass gain after three weeks. In fact, the actual experiment lasted 8 weeks and examined many additional variables. The data referenced here are available in LightatNight8weeks. Some of the variables are summarized in the following table. Variable Name Condition BMGain Corticosterone DayPct Variable Description Treatment group (Light or Dark) Change in body mass (in grams) after 8 weeks Blood corticosterone level (a measure of stress) Percentage of calories consumed during the day Average daily food consumption (in grams) Glucose intolerant at end of study? (Yes or No) A measure of physical activity level Consumption Glocoselnt Activity 1. What are the cases? What is the explanatory variable? Is this a randomized experiment or an observational study? 2. What is the mean body mass gain for all the mice in this dataset? What is the five number summary? Show a graph of the body mass gains in the sample. Do there appear to be any outliers? Find and interpret a 95% confidence interval for mean body mass gain. ## [1] 8.617647 The mean increase in body mass is 8.618 grams. The five number summary is: ## 0% 25% 50% 75% 100% ## 2.79 6.34 9.05 9.89 17.40 5- 4 2 0- 10 BMGain 5 15 To find a 95% confidence interval, we need to use the bootstrap method 500 400 300- 200 100- 10 12 Bootstrap sample Body Mass Gains (in grams) Since this distribution is relatively symmetric and bell-shaped, we can use the 95% rule Since this distribution is relatively symmetric and bell-shaped, we can use the 95% rule. ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## Based on 10000 bootstrap replicates ## CALL : ## boot.ci(boot.out gains-boot, type "basic") = _ ## Intervals : ## Level ## 95% ( 7.012, 1.151 ) ## Calculations and Intervals on Original Scale Basic 3. The primary question in the study was whether having a light on at night causes greater average body mass gain. Give side-by-side boxplots comparing body mass gain for the two groups. Does having a light on at night appear to increase body mass gain in mice? Light- Dark 15 10 BMGain 4. Conduct a hypothesis test to see if the data provide evidence that mean body mass gain is higher for mice with a light on at night. Show all details of the test. If the results are significant, can we conclude that there is a causal association? Why or why not? 5. To examine how large is the effect on body mass gain of having a light on at night, give a 99% confidence interval for the difference in mean body mass gain between mice with a light on at night and mice with darkness at night. Interpret the interval in context. 6. Perhaps the difference in weight gain between the two groups was affected by a difference in food consumption. Conduct a hypothesis test for a difference in mean consumption between the two groups. Is there evidence of a significant difference in average food consumption betweern the two groups? 7. Perhaps the difference in weight gain between the two groups was affected by a difference in activity level. Conduct a hypothesis test for a difference in mean activity level between the two groups. Is there evidence of a significant difference in average activity level between the two groups? 8. Perhaps the difference in weight gain between the two groups was affected by a difference in the time of day in which the mice at their food. Conduct a hypothesis test for a difference in mean percentage of calories consumed during the day between the two groups. Is there evidence of a significant difference in mean percentage between the two groups? (Mice in the wild eat almost all of their calories during the dark nighttime hours, so a high percentage eaten during the day is unusual for mice.) 9. Summarize the results of the tests in parts (6), (7), and (8). For any that show a significant effect, conduct a test to determine if there is a significant correlation between that quantitative variable and body mass gain. Is the correlation positive or negative, and what does that mean in this context? 10. The study examined other effects of light at night in addition to body mass gain. Conduct a test to see if the proportion of mice who become glucose intolerant is higher for mice with a light on than mice in darkness. Note that in StatKey all you need is the relevant counts and sample sizes (not the full dataset) to run this test. 11. Higher blood corticosterone levels indicate greater physiological levels of stress. Find and interpret a 90% confidence interval for the difference in mean corticosterone levels between mice with a light on and mice in darkness. Does the confidence interval allow us to conclude that theree is a difference in mean corticosterone levels between the two groups? If so, which group has higher mean levels? 12. Write a paragraph summaring the results of your findings. If the results of the study can be generalized to humans, what are the take-away messages?

Answer 1

1)There are two cases Dark and Light. These two are predictor variables and BMI is the explanatory variable. It's a randomized experiment as we are doing an experiment on mice using an input variable which is the presence of light or not.

2 8 6 4 2 5

4)

> mean(light)
[1] 6.732
> mean(dark)
[1] 4.11375
> sd(light)
[1] 2.965733
> sd(dark)

> length(light)
[1] 10
> length(dark)
[1] 8

The provided sample means are shown below: X 6.732 X2 4.11375 Also, the provided sample standard deviations are: 812.965733 89 1.557269 and the sample sizes are n-10 and n-= 8. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used

Based on the information provided, the significance level is a 0.05, and the degrees of freedom are df- 16. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal Hence, it is found that the critical value for this two-tailed test is te 2.12, for a 0.05 and df 16. The rejection region for this two-tailed test is R-{t: t> 2.12} ()taisic; Since it is assumed that the population variances are equal, the t-statistic is computed as follows: n1+n2-2 6.732 4.11375 2.252 20 7018872001) (4) Decision about the null hypothesis Since it is observed that t| rejected 2.252 > tcー2.12, ît is then concluded that the null hypothesis is

Using the P-value approach: The p-value is p 0.0387, and since p-0.0387<0.05, it is concluded that the null hypothesis is rejected. 5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1 is different than po, at the 0.05 significance level. Confidence Interval The 95% confidence interval is 0.153 < 1- 2 < 5.083.

Confidence Interval The 99% confidence interval is 0.778 < μ1-Ha < 6.014.