Question

Determine the appropriate critical value(s) for each of the following tests concerning the population mean: a. upper-tailed test: a=0.05; n=36; o=5.0 b. lower-tailed test: a=0.01; n=30; s=6.0 c. two-tailed test: a=0.02; n=31; s=5.1 d. two-tailed test: a=0.10; n=25; o=5.7 a. Determine the appropriate critical value for an upper-tailed test: a=0.05; n=36; a=5.0. The critical value(s) is (are) (Round to two decimal places as needed. Use a comma to separate answers as needed.)

Answer 1

To determine the appropriate critical values for each of the tests concerning the population mean, we can use the t-distribution table or a statistical calculator. The critical value for each test depends on the significance level (alpha) and the degrees of freedom (df), which are related to the sample size.

a. Upper-tailed test:

- Alpha (α) = 0.05

- Sample size (n) = 36

- Standard deviation (σ) = 5.0

To find the critical value for this upper-tailed test, we need to calculate the degrees of freedom (df), which is (n - 1) in this case:

df = 36 - 1 = 35

Now, we can find the critical value from the t-distribution table or use a calculator. For a one-tailed test with alpha = 0.05 and df = 35, the critical value is approximately 1.686.

So, the critical value for this upper-tailed test is approximately 1.686 (rounded to two decimal places).

b. Lower-tailed test:

- Alpha (α) = 0.01

- Sample size (n) = 30

- Sample standard deviation (s) = 6.0

For a lower-tailed test, we can still calculate the degrees of freedom:

df = 30 - 1 = 29

Now, we can find the critical value for this lower-tailed test with alpha = 0.01 and df = 29 using the t-distribution table or a calculator. The critical value is approximately -2.621.

So, the critical value for this lower-tailed test is approximately -2.621 (rounded to two decimal places).

c. Two-tailed test:

- Alpha (α) = 0.02

- Sample size (n) = 31

- Sample standard deviation (s) = 5.1

For a two-tailed test, we need to divide the alpha level by 2 to account for both tails:

Alpha (α/2) = 0.02 / 2 = 0.01

Now, calculate the degrees of freedom:

df = 31 - 1 = 30

Using the t-distribution table or a calculator with alpha = 0.01 and df = 30, you can find the critical values. The critical values are approximately ±2.750.

So, the critical values for this two-tailed test are approximately ±2.750 (rounded to two decimal places).

d. Two-tailed test:

- Alpha (α) = 0.10

- Sample size (n) = 25

- Standard deviation (σ) = 5.7

For a two-tailed test, we can calculate the degrees of freedom:

df = 25 - 1 = 24

Now, using the t-distribution table or a calculator with alpha = 0.10 and df = 24, you can find the critical values. The critical values are approximately ±1.711.

So, the critical values for this two-tailed test are approximately ±1.711 (rounded to two decimal places).

Explanation: