Question

Is a t-distribution Appropriate? We give summary statistics and a dotplot for a sample. Indicate whether or not it is appropriate to use the t-distribution. If it is appropriate, give the degrees of freedom for the t-distribution and give estimated standard error. It it's not appropriate, input 0 for both the degrees of freedom and the estimated standard error.

A sample with n = 150, x⎯⎯=49.5x¯=49.5, and s = 10.6

not appropriateappropriate
degrees of freedom =
SE = (round to three decimal places)

Answer 1

It is appropriate to use the t-distribution for the given sample with n = 150. The degrees of freedom would be 149 and the estimated standard error is calculated to be approximately 0.865 when rounded to three decimal places.

Step-by-step explanation:

To determine if it is appropriate to use the t-distribution for the given sample, consider the sample size and the lack of the population standard deviation. In this case, the sample size is n = 150, which is much larger than the rule-of-thumb cutoff of 30 for small sample sizes. Since the population standard deviation is not given, this is a perfect scenario to use the sample standard deviation s as an estimate for the unknown population standard deviation.

The appropriate degrees of freedom for the t-distribution are calculated as df = n - 1. Therefore, df = 150 - 1 = 149. The estimated standard error (SE) is calculated using the formula SE = s / sqrt(n), which gives us SE = 10.6 / sqrt(150).

The error bound for a population mean using the t-distribution is more suited here than the normal distribution, especially because the population standard deviation is not known and 's' is being used as the estimate.

In conclusion, it is appropriate to use the t-distribution with degrees of freedom = 149. The estimated standard error is SE = 0.865 (rounded to three decimal places).