Question

t time When constructing confidence intervals for a population mean, we almost always use critical values from a t distribution rather than the standard Normal distribution.

a. When is it necessary to use a t critical value rather than a z critical value when constructing a confidence interval for a population means?

b. For a particular level of confidence, explain what happens to the t critical values as Page Number: 548 the degrees of freedom increase.

Short answer

Part a) When the sample standard deviation is known and the sample size is less than$30$, the $t$-distribution is used to calculate the critical value.

Part b) It could be said that as the degree of freedom increases, the $t$critical value approaches the $z$critical value.

Step-by-step solution

Part a) Step 1: The objective is to explain the condition required to construct the confidence interval using the t critical value rather than z

When the standard deviation is known, the Z-distribution is used to compute the confidence interval for the population mean.

When the sample standard deviation is known and the sample size is less than$30$, the$t$-distribution is used to calculate the critical value.

Part b) Step 2: The objective is to explain the effect the increase of the degree of freedom on tcritical value for a particular confidence level.

One of the t-properties distributions is that as the degree of freedom increases, the $t$-distribution approaches the normal distribution.

Therefore, It could be said that as the degree of freedom increases, the $t$critical value approaches the $z$critical value.