Assume that we want to construct a confidence interval. Do...

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value $t_{\alpha / 2}$ (b) find the critical value $z_{\alpha / 2},$ or $(c)$ state that neither the normal distribution nor the $t$ distribution applies. Confidence level is $90 \%, \sigma$ is not known, and the histogram of 61 player salaries (thousands of dollars) is as shown.

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value $t_{\alpha / 2}$
(b) find the critical value $z_{\alpha / 2},$ or $(c)$ state that neither the normal distribution nor the $t$ distribution applies.
Confidence level is $90 \%, \sigma$ is not known, and the histogram of
61 player salaries (thousands of dollars) is as shown.

Key Concepts

Confidence Interval

A confidence interval is a range of values constructed from sample data, used to estimate an unknown population parameter. It reflects the uncertainty inherent in the sampling process, with the confidence level indicating the frequency at which these intervals would contain the true parameter over many repetitions.

t Distribution

The t distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is relatively small or the estimation of the standard deviation introduces extra uncertainty. It is more spread out than the normal distribution, and its exact shape depends on the degrees of freedom from the sample size.

Degrees of Freedom

Degrees of freedom, often calculated as the sample size minus one when estimating a mean, determine the exact form of the t distribution. They reflect the amount of independent information available for estimation and influence the critical values used in constructing confidence intervals.

Critical Value

A critical value is the cutoff point that defines the margin of error for a confidence interval. It is chosen from the appropriate distribution (either z for known variance or t for unknown variance) for the desired confidence level, such as 90%, and marks the point beyond which a certain percentage of the probability distribution lies.

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