Question

Find the rejection region for each test below, using the given information for each test. If n is small, you may assume that the original distribution is normal. You do NOT need to carry out the full test.

c) x ̄ = 10

n=8

s=4

α = 0.1

Ho :μ=12

Ha : μ≠12

Answer 1

Solution:

From given information, it is observed that the test is two tailed test.

α = 0.1

n = 8

df = n - 1 = 8 - 1 = 7

So, critical t value by using t-table is given as below:

Critical t value = ±1.895

Rejection region: Reject H₀ if t < -1.895 or t > 1.895.

Answer 2

To find the rejection region for a two-tailed t-test with the given information, we first need to calculate the critical t-value(s) that correspond to the significance level (α = 0.1) and the degrees of freedom (df = n - 1).

Given information: Sample mean (x̄) = 10 Sample size (n) = 8 Sample standard deviation (s) = 4 Significance level (α) = 0.1 Null hypothesis (Ho): μ = 12 Alternative hypothesis (Ha): μ ≠ 12

Step 1: Calculate the degrees of freedom (df) df = n - 1 df = 8 - 1 df = 7

Step 2: Find the critical t-value(s) for a two-tailed test at α = 0.1 and df = 7. Using a t-table or a statistical software, we find the critical t-values as follows: t-critical (lower tail) = -1.895 t-critical (upper tail) = 1.895

Step 3: Determine the rejection region. For a two-tailed test, the rejection region consists of extreme values in both tails of the t-distribution. Any t-value that falls beyond the critical t-values obtained in Step 2 will lead to the rejection of the null hypothesis.

Rejection region: t < -1.895 or t > 1.895

If the calculated t-value (t) from the sample falls outside this rejection region, we would reject the null hypothesis and conclude that there is enough evidence to support the alternative hypothesis, suggesting that the population mean is different from 12. However, if the calculated t-value falls within the rejection region, we would fail to reject the null hypothesis and do not have sufficient evidence to conclude that the population mean is different from 12 at the 0.1 significance level.