Data 4.1 on page 258 introduces a study in which mice that...

Data 4.1 on page 258 introduces a study in which mice that had a light on at night (rather than complete darkness) ate most of their calories when they should have been resting. These mice gained a significant amount of weight, despite eating the same number of calories as mice kept in total darkness. The time of eating seemed to have a significant effect. Exercise 6.96 examines the mice with dim light at night. A second group of mice had bright light on all the time (day and night). There were nine mice in the group with bright light at night and, after 8 weeks, they gained an average of $11.0 \mathrm{~g}$ with a standard deviation of $2.6 .$ The data are shown in Figure $6.10 .$ Is it appropriate to use a t-distribution in this situation? Why or why not? If not, how else might we construct a confidence interval for mean weight gain of mice with a bright light on all the time?

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Key Concepts

Confidence intervals

Confidence intervals provide a range of plausible values for a population parameter such as the mean. Their construction often relies on specific distributional assumptions (like normality) or large-sample approximations, and choosing the appropriate method hinges on meeting these underlying assumptions.

Bootstrap methods

Bootstrap methods are non-parametric techniques that involve resampling the data to estimate the sampling distribution of a statistic. They are particularly useful when the assumptions required for parametric methods, such as normality in the t-distribution, are questionable or when the sample size is small.

t-distribution

The t-distribution is used when making inferences about means when the sample size is small and the population standard deviation is unknown. It becomes a good approximation to the true sampling distribution of the sample mean particularly when the underlying data distribution is normal or nearly normal.

Normality assumption

A primary assumption for the valid use of the t-distribution is that the data are drawn from a normally distributed population. With small sample sizes, this assumption is critical because the central limit theorem does not ensure normality of the sample mean when the sample size is too small.

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