Answer :
To find the critical value(s) for the given hypotheses tests, we follow these steps:
1. Test 1: [tex]\( H_a: \mu < 65, \alpha = 0.05, n = 27 \)[/tex]
- The hypothesis test is left-tailed (since [tex]\( H_a \)[/tex] specifies [tex]\(\mu < 65\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.05\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(27\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(26\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a left-tailed test:
- The critical value corresponds to the [tex]\(5\%\)[/tex] (since [tex]\(\alpha = 0.05\)[/tex]) left tail of the t-distribution with [tex]\(26\)[/tex] degrees of freedom
The critical value is approximately [tex]\( -1.7056 \)[/tex].
So, the critical value for test 1 is: -1.7056
2. Test 2: [tex]\( H_a: \mu \neq 200, n = 15, \alpha = 0.10 \)[/tex]
- The hypothesis test is two-tailed (since [tex]\(H_a\)[/tex] specifies [tex]\(\mu \neq 200\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.10\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(15\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(14\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a two-tailed test:
- The critical values correspond to the [tex]\(5\%\)[/tex] left tail and [tex]\(5\%\)[/tex] right tail of the t-distribution with [tex]\(14\)[/tex] degrees of freedom (since [tex]\(\alpha = 0.10\)[/tex], we split it into [tex]\(0.05\)[/tex] for each tail).
The critical values are approximately [tex]\( \pm 1.7613 \)[/tex].
So, the critical values for test 2 are: -1.7613, 1.7613
3. Test 3: [tex]\( H_o: \mu \leq 15.26, n = 35, \bar{x} =16.25, \sigma = 1.25, \alpha = 0.05 \)[/tex]
- The hypothesis test is right-tailed (since [tex]\(H_o\)[/tex] specifies [tex]\(\mu \leq 15.26\)[/tex] and we are likely testing [tex]\(H_a: \mu > 15.26\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.05\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(35\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) are [tex]\(34\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a right-tailed test:
- The critical value corresponds to the [tex]\(95\%\)[/tex] (since [tex]\(1 - \alpha = 0.95\)[/tex]) right tail of the t-distribution with [tex]\(34\)[/tex] degrees of freedom.
The critical value is approximately [tex]\(1.6909\)[/tex].
So, the critical value for test 3 is: 1.6909
4. Test 4: [tex]\( H_a: \mu \neq 80, \bar{x} = 75.4, \sigma = 19.05, \alpha = 0.01, n = 50 \)[/tex]
- The hypothesis test is two-tailed (since [tex]\(H_a\)[/tex] specifies [tex]\(\mu \neq 80\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.01\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(50\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) are [tex]\(49\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a two-tailed test:
- The critical values correspond to the [tex]\(0.5\%\)[/tex] left tail and [tex]\(0.5\%\)[/tex] right tail of the t-distribution with [tex]\(49\)[/tex] degrees of freedom (since [tex]\(\alpha = 0.01\)[/tex], we split it into [tex]\(0.005\)[/tex] for each tail).
The critical values are approximately [tex]\( \pm 2.6800 \)[/tex].
So, the critical values for test 4 are: -2.6800, 2.6800
In summary:
1. -1.7056
2. -1.7613, 1.7613
3. 1.6909
4. -2.6800, 2.6800
1. Test 1: [tex]\( H_a: \mu < 65, \alpha = 0.05, n = 27 \)[/tex]
- The hypothesis test is left-tailed (since [tex]\( H_a \)[/tex] specifies [tex]\(\mu < 65\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.05\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(27\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(26\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a left-tailed test:
- The critical value corresponds to the [tex]\(5\%\)[/tex] (since [tex]\(\alpha = 0.05\)[/tex]) left tail of the t-distribution with [tex]\(26\)[/tex] degrees of freedom
The critical value is approximately [tex]\( -1.7056 \)[/tex].
So, the critical value for test 1 is: -1.7056
2. Test 2: [tex]\( H_a: \mu \neq 200, n = 15, \alpha = 0.10 \)[/tex]
- The hypothesis test is two-tailed (since [tex]\(H_a\)[/tex] specifies [tex]\(\mu \neq 200\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.10\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(15\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(14\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a two-tailed test:
- The critical values correspond to the [tex]\(5\%\)[/tex] left tail and [tex]\(5\%\)[/tex] right tail of the t-distribution with [tex]\(14\)[/tex] degrees of freedom (since [tex]\(\alpha = 0.10\)[/tex], we split it into [tex]\(0.05\)[/tex] for each tail).
The critical values are approximately [tex]\( \pm 1.7613 \)[/tex].
So, the critical values for test 2 are: -1.7613, 1.7613
3. Test 3: [tex]\( H_o: \mu \leq 15.26, n = 35, \bar{x} =16.25, \sigma = 1.25, \alpha = 0.05 \)[/tex]
- The hypothesis test is right-tailed (since [tex]\(H_o\)[/tex] specifies [tex]\(\mu \leq 15.26\)[/tex] and we are likely testing [tex]\(H_a: \mu > 15.26\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.05\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(35\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) are [tex]\(34\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a right-tailed test:
- The critical value corresponds to the [tex]\(95\%\)[/tex] (since [tex]\(1 - \alpha = 0.95\)[/tex]) right tail of the t-distribution with [tex]\(34\)[/tex] degrees of freedom.
The critical value is approximately [tex]\(1.6909\)[/tex].
So, the critical value for test 3 is: 1.6909
4. Test 4: [tex]\( H_a: \mu \neq 80, \bar{x} = 75.4, \sigma = 19.05, \alpha = 0.01, n = 50 \)[/tex]
- The hypothesis test is two-tailed (since [tex]\(H_a\)[/tex] specifies [tex]\(\mu \neq 80\)[/tex]).
- The significance level ([tex]\(\alpha\)[/tex]) is [tex]\(0.01\)[/tex].
- The sample size ([tex]\(n\)[/tex]) is [tex]\(50\)[/tex], so the degrees of freedom ([tex]\(df\)[/tex]) are [tex]\(49\)[/tex] (which is [tex]\(n - 1\)[/tex]).
For a two-tailed test:
- The critical values correspond to the [tex]\(0.5\%\)[/tex] left tail and [tex]\(0.5\%\)[/tex] right tail of the t-distribution with [tex]\(49\)[/tex] degrees of freedom (since [tex]\(\alpha = 0.01\)[/tex], we split it into [tex]\(0.005\)[/tex] for each tail).
The critical values are approximately [tex]\( \pm 2.6800 \)[/tex].
So, the critical values for test 4 are: -2.6800, 2.6800
In summary:
1. -1.7056
2. -1.7613, 1.7613
3. 1.6909
4. -2.6800, 2.6800
