If we use the sample data in Exercise 2 for constructing a...

If we use the sample data in Exercise 2 for constructing a $99 \%$ confidence interval, what is the number of degrees of freedom that should be used for finding the critical value of $t_{\alpha / 2}$ ? What is the critical value $t_{\alpha / 2} ?$

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Key Concepts

Degrees of Freedom

When constructing a confidence interval using the t-distribution, the degrees of freedom (df) are determined by the sample size. Specifically, if the sample size is n, then the number of degrees of freedom is n - 1. This adjustment accounts for the fact that one parameter – the mean – is estimated from the data.

t-Distribution

The t-distribution is a probability distribution used in situations where the population standard deviation is unknown and the sample size is relatively small. It is similar to the normal distribution but has heavier tails, which provides a more conservative estimate when constructing confidence intervals.

Critical Value

The critical value t?/2 for a given confidence level (in this case 99%) is the cutoff point from the t-distribution that leaves an area of ?/2 (where ? = 1 - confidence level, so ?/2 = 0.005 for a 99% confidence interval) in each tail of the distribution. This value depends on both the chosen confidence level and the degrees of freedom.

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