Question

In order to construct a confidence interval for the population variance, a random sample of n observations is drawn from a normal population. Use this information to find χ²_α/2,df and χ²_1-α/2,df under the following scenarios. (Round your answers to 3 decimal places. You may find it useful to reference the appropriate table: chi-square table or F table)

		χ2α/2,df	χ21-α/2,df
a.	A 90% confidence level with n = 20.
b.	A 90% confidence level with n = 43.
c.	A 99% confidence level with n = 20.
d.	A 99% confidence level with n = 43.

Answer 1

SOLUTION:

From given data,

In order to construct a confidence interval for the population variance, a random sample of n observations is drawn from a normal population. Use this information to find χ²_α/2,df and  χ²_1-α/2,df under the following scenarios.

(a) A 90% confidence level with n = 20

90% = 90/100 = 0.90

= 1-0.90 = 0.1

/2 = 0.1/2 = 0.05

1-/2 = 1-0.05 = 0.95

Degree of freedom = df = n-1 = 20-1 = 19

χ²_α/2,df =  χ²_0.05,19 = 30.14

χ²_1-α/2,df = χ²_0.95,19 = 10.12

(b) A 90% confidence level with n = 43

90% = 90/100 = 0.90

= 1-0.90 = 0.1

/2 = 0.1/2 = 0.05

1-/2 = 1-0.05 = 0.95

Degree of freedom = df = n-1 = 43-1 = 42

χ²_α/2,df =  χ²_0.05,42 = 58.12

χ²_1-α/2,df = χ²_0.95,42 = 28.14

(c) A 99% confidence level with n = 20

99% = 99/100 = 0.99

= 1-0.99 = 0.01

/2 = 0.01/2 = 0.005

1-/2 = 1-0.005 = 0.995

Degree of freedom = df = n-1 = 20-1 = 19

χ²_α/2,df =  χ²_0.005,19= 38.582

χ²_1-α/2,df = χ²_0.995,19 = 6.844

(d) A 99% confidence level with n = 43.

99% = 99/100 = 0.99

= 1-0.99 = 0.01

/2 = 0.01/2 = 0.005

1-/2 = 1-0.005 = 0.995

Degree of freedom = df = n-1 = 43-1 = 42

χ²_α/2,df =  χ²_0.005,42= 68.053

χ²_1-α/2,df = χ²_0.995,42 = 22.138

Values of the Chi-squared distribution