Determine the critical region and the critical values used to test the following null hypotheses:
a. Ho: μ = 55 (≥), Ha: μ < 55, α = 0.02
b. Ho: μ = −86 (≥), Ha: μ < −86, α = 0.01
c. Ho: μ = 107, Ha: μ ≠ 107, α = 0.05
d. Ho: μ = 17.4 (≤), Ha: μ > 17.4, α = 0.10
To determine the critical region and critical values for hypothesis testing, we first need to identify the type of test (one-tailed or two-tailed) and the significance level (α).
a. For the null hypothesis Ho: μ = 55 (≥) and alternative hypothesis Ha: μ < 55, with α = 0.02, this is a one-tailed test to the left. To find the critical region, we need to find the Z-score that corresponds to the desired significance level. Since it is a one-tailed test to the left, we need to find the Z-score that has an area of 0.02 to its left. From a standard normal distribution table or calculator, we find this to be approximately -2.05. Therefore, the critical value is -2.05.
b. For the null hypothesis Ho: μ = -86 (≥) and alternative hypothesis Ha: μ < -86, with α = 0.01, this is again a one-tailed test to the left. To find the critical region, we need to find the Z-score that corresponds to the desired significance level. Since it is a one-tailed test to the left, we need to find the Z-score that has an area of 0.01 to its left. From a standard normal distribution table or calculator, we find this to be approximately -2.33. Therefore, the critical value is -2.33.
c. For the null hypothesis Ho: μ = 107 and alternative hypothesis Ha: μ ≠ 107, with α = 0.05, this is a two-tailed test. The significance level needs to be split equally into two tails. Each tail will have an area of α/2 = 0.05/2 = 0.025. To find the critical region, we need to find the Z-scores that correspond to the upper and lower tail areas. From a standard normal distribution table or calculator, we find the critical values for the upper and lower tails to be approximately ±1.96. Therefore, the critical values are -1.96 and 1.96.
d. For the null hypothesis Ho: μ = 17.4 (≤) and alternative hypothesis Ha: μ > 17.4, with α = 0.10, this is a one-tailed test to the right. To find the critical region, we need to find the Z-score that corresponds to the desired significance level. Since it is a one-tailed test to the right, we need to find the Z-score that has an area of 0.10 to its right. From a standard normal distribution table or calculator, we find this to be approximately 1.28. Therefore, the critical value is 1.28.
In summary:
a. Critical region: Z < -2.05
Critical value: -2.05
b. Critical region: Z < -2.33
Critical value: -2.33
c. Critical region: Z < -1.96 or Z > 1.96
Critical values: -1.96 and 1.96
d. Critical region: Z > 1.28
Critical value: 1.28
To determine the critical region and the critical values for hypothesis testing, we need to consult the appropriate statistical table based on the specific significance level (α) and the type of test (one-tailed or two-tailed). The critical region is the area in the tails of the distribution where we would reject the null hypothesis, and the critical values are the specific test statistics or threshold values that define this region.
a. For a one-tailed test where Ha: μ < 55 and α = 0.02, the critical region is in the left tail of the distribution. To determine the critical value, we need to find the z-score corresponding to the area under the left tail equal to α = 0.02. By referring to the standard normal distribution table, we find that the critical value for this test is approximately -2.05.
b. For a one-tailed test where Ha: μ < -86 and α = 0.01, the critical region is in the left tail of the distribution. By referring to the standard normal distribution table, we find that the critical value for this test is approximately -2.33.
c. For a two-tailed test where Ha: μ ≠ 107 and α = 0.05, the critical region is divided between the left and right tails of the distribution. To determine the critical values, we need to find the z-scores corresponding to the area under each tail equal to α/2 = 0.05/ 2 = 0.025. By referring to the standard normal distribution table, we find that the critical values for this test are approximately -1.96 and 1.96.
d. For a one-tailed test where Ha: μ > 17.4 and α = 0.10, the critical region is in the right tail of the distribution. By referring to the standard normal distribution table, we find that the critical value for this test is approximately 1.28.
In summary:
a. Critical region: Z < -2.05
b. Critical region: Z < -2.33
c. Critical region: Z < -1.96 or Z > 1.96
d. Critical region: Z > 1.28