Using Correct Distribution.Assume that we want to construct...

Using Correct Distribution.Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value t ? / 2 , (b) find the critical value z ? / 2 , (c) state that neither the normal distribution nor the t distribution applies. Confidence level is 95%, ? is known to be $4,385,000, and the dotplot of a sample of Red Sox baseball player salaries is as shown below.

Using Correct Distribution.Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value t ? / 2 , (b) find the critical value z ? / 2 , (c) state that neither the normal distribution nor the t distribution applies.
Confidence level is 95%, ? is known to be $4,385,000, and the dotplot of a sample of Red Sox baseball player salaries is as shown below.

Key Concepts

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true value of an unknown population parameter. It is constructed using the point estimate, a margin of error, and the appropriate critical value from a probability distribution based on the sampling distribution of the estimator.

Critical Value

A critical value is a point on a probability distribution that is used to determine the margin of error in a confidence interval. In the context of confidence intervals, it corresponds to the desired level of confidence and represents the number of standard deviations (or standard errors) away from the mean needed to capture the central portion of the distribution.

Normal (z) Distribution

The normal distribution, or z distribution, is used when the population standard deviation is known and either the sample size is large or the population is normally distributed. When these conditions hold, the sampling distribution of the sample mean is approximately normal, allowing for the use of z critical values for constructing confidence intervals.

t Distribution

The t distribution is used when the population standard deviation is unknown and is estimated using the sample standard deviation, especially with small sample sizes. It has heavier tails than the normal distribution, which accounts for the extra variability due to the estimation of the standard deviation, and the appropriate critical values are determined based on degrees of freedom.

Distribution Assumptions and Data Assessment

Selecting the correct distribution for constructing a confidence interval requires assessing the underlying assumptions about the data. This includes examining whether the population is approximately normally distributed (or if the sample size is sufficiently large for the central limit theorem to apply), whether the population standard deviation is known, and evaluating any potential skewness or outliers in the data distribution (for example, via a dotplot). These factors help determine whether the normal distribution, the t distribution, or another approach is appropriate.

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