Question

We use the t distribution to construct a confidence interval for the population mean when the underlying population standard deviation is not known. Under the assumption that the population is normally distributed, find tα/2,df for the following scenarios. (You may find it useful to reference the t table. Round your answers to 3 decimal places.) tα/2,df a. A 90% confidence level and a sample of 7 observations. b. A 95% confidence level and a sample of 7 observations. c. A 90% confidence level and a sample of 29 observations. d. A 95% confidence level and a sample of 29 observations.

Answer 1

Solution :

Given that,

(a)

sample size = n = 7

Degrees of freedom = df = n - 1 = 7 - 1 = 6

At 90% confidence level the t is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_{0.05, 6} = 1.643

(b)

sample size = n = 7

Degrees of freedom = df = n - 1 = 7 - 1 = 6

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_{0.05 , 6} = 2.447

(c)

sample size = n = 29

Degrees of freedom = df = n - 1 = 29 -1 = 28

At 90% confidence level the t is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_{0.05, 28} = 1.701

(d)

sample size = n = 29

Degrees of freedom = df = n - 1 = 29 - 1 = 28

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_{0.05 , 28} = 2.048