Question

Confidence interval when sigma is unknown.

2. Find the critical value tα2 for the given level and sample size:

1. Level 90%, Sample size is 13,

df =                , so the critical value is: =                      

2. Level 98%, Sample size is 10

df =                , so the critical value is: =                      

3. Level 80%, Sample size is 17

df =                , so the critical value is: =                      

4. Level 99%, Sample size is 77

df =                , so the critical value is: =                      

Answer 1

a) Given that, level 90%, sample size is 13

=> significance level = 1 - 0.90 = 0.10

=> Degrees of freedom = 13 - 1 = 12

Using Excel we get, t-critical value at significance level of 0.10 with 12 degrees of freedom is tα/2 = 1.782

Excel Command : =TINV(0.10, 12) = 1.782

Therefore, df = 12, so the critical value = 1.782

b) Given that, level 98%, sample size is 10

=> significance level = 1 - 0.98 = 0.02

=> Degrees of freedom = 10 - 1 = 9

Using Excel we get, t-critical value at significance level of 0.02 with 9 degrees of freedom is tα/2 = 2.821

Excel Command : =TINV(0.02, 9) = 2.821

Therefore, df = 9, so the critical value = 2.821

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c) Given that, level 80%, sample size is 17

=> significance level = 1 - 0.80 = 0.20

=> Degrees of freedom = 17 - 1 = 16

Using Excel we get, t-critical value at significance level of 0.20 with 16 degrees of freedom is tα/2 = 1.337

Excel Command : =TINV(0.20, 16) = 1.337

Therefore, df = 16, so the critical value = 1.337

d) Given that, level 99%, sample size is 77

=> significance level = 1 - 0.99 = 0.01

=> Degrees of freedom = 77 - 1 = 76

Using Excel we get, t-critical value at significance level of 0.01 with 76 degrees of freedom is tα/2 = 2.642

Excel Command : =TINV(0.01, 76) = 2.642

Therefore, df = 76, so the critical value = 2.642