Question

A more extensive tabulation of t critical values than what appears in this book shows that for the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20, and .15, respectively. What is the confidence level for each of the following three confidence intervals for the mean m of a normal population distribution

Answer 1

a) the interval has confidence level of 70%

b) the interval has confidence level of 70%

c) the interval has confidence level of 70%

Explanation:

Given that;

the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20, and .15, respectively

a)

{ X - 0.687 √(s/21), X + 1.725 √(s/21) }

it given that the area to the right of 0.687 is 0.25

so from the table, the area to the right of 1.725 is 0.05

Sum the areas that are outside of the interval will be

0.05 + 0.25 = 0.30

so (1 - 0.30) = 0.7 = 70%

therefore the interval has confidence level of 70%

b)

{ X - 0.860 √(s/21), X + 1.325 √(s/21) }

given area to the right of 0.860 is 0.2

so from the table, the area to the right of 1.325 is 0.1

Sum the areas that are outside of the interval will be

0.2 + 0.1 = 0.3

so (1 - 0.3) = 0.7 = 70%

therefore the interval has confidence level of 70%

c)

{ X - 1.064 √(s/21), X + 1.064 √(s/21) }

given area to the right of 1.064 is 0.15,

the interval is symmetric

therefore sum of two areas that are outside the interval is;

0.15 + 0.15 = 0.30

so (1 - 0.30) = 0.7 = 70%

therefore the interval has confidence level of 70%