Answer :
Sure! Let's go through each part of the problem step-by-step to understand the solution.
### Part (a)
Hypothesis Test:
We need to test the null hypothesis [tex]\( H_0: \mu = 100 \)[/tex] against the alternative hypothesis [tex]\( H_1: \mu < 100 \)[/tex]. This is a left-tailed test.
1. Sample Information:
- Sample size ([tex]\( n \)[/tex]) = 8
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 100.2
- Population standard deviation ([tex]\( \sigma \)[/tex]) = 2
2. Calculate the z-score:
The z-score is calculated using the formula:
[tex]\[
z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}
\][/tex]
Substituting the values:
[tex]\[
z = \frac{100.2 - 100}{2/\sqrt{8}} = 0.2828
\][/tex]
3. Determine the critical value:
For a significance level ([tex]\( \alpha \)[/tex]) of 0.05, the critical z-value for a left-tailed test is approximately -1.645.
4. Decision Rule:
If the calculated z-score is less than the critical value, we reject the null hypothesis. In this case, [tex]\( z = 0.2828 \)[/tex] is not less than [tex]\(-1.645\)[/tex].
5. Conclusion:
Since the z-score is not lower than the critical value, we do not reject the null hypothesis. There is not enough evidence to suggest that the mean speed is less than 100 meters per second.
### Part (b)
Power of the Test:
We want to calculate the power of the test under the condition that the true mean speed is 95 meters per second.
1. True mean ([tex]\( \mu_1 \)[/tex]): 95
2. Calculate z-value for [tex]\( \mu_1 \)[/tex]:
The power of the test is given by the area to the right of the critical point for the alternative distribution.
[tex]\[
z_{\beta} = \frac{\mu_1 - \mu_0}{\sigma/\sqrt{n}} - z_{\text{critical}}
\][/tex]
3. Power Calculation:
Using the given information, the power of the test is 0.999999971.
This means that the test is almost certain to correctly reject the null hypothesis if the true mean speed is 95 meters per second.
### Part (c)
Sample Size for Desired Power:
We need to determine the sample size required to detect a mean speed as low as 93 meters per second with a power of 0.85.
1. True mean ([tex]\( \mu_1 \)[/tex]): 93
2. Desired power: 0.85
3. Calculate the required sample size:
Using the information provided, after performing the necessary statistical calculations, the required sample size is 1.
This implies that even a small sample size is sufficient to achieve a power of at least 0.85 when detecting a mean speed as low as 93 meters per second.
This completes the detailed breakdown of the solution for the given problem.
### Part (a)
Hypothesis Test:
We need to test the null hypothesis [tex]\( H_0: \mu = 100 \)[/tex] against the alternative hypothesis [tex]\( H_1: \mu < 100 \)[/tex]. This is a left-tailed test.
1. Sample Information:
- Sample size ([tex]\( n \)[/tex]) = 8
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 100.2
- Population standard deviation ([tex]\( \sigma \)[/tex]) = 2
2. Calculate the z-score:
The z-score is calculated using the formula:
[tex]\[
z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}
\][/tex]
Substituting the values:
[tex]\[
z = \frac{100.2 - 100}{2/\sqrt{8}} = 0.2828
\][/tex]
3. Determine the critical value:
For a significance level ([tex]\( \alpha \)[/tex]) of 0.05, the critical z-value for a left-tailed test is approximately -1.645.
4. Decision Rule:
If the calculated z-score is less than the critical value, we reject the null hypothesis. In this case, [tex]\( z = 0.2828 \)[/tex] is not less than [tex]\(-1.645\)[/tex].
5. Conclusion:
Since the z-score is not lower than the critical value, we do not reject the null hypothesis. There is not enough evidence to suggest that the mean speed is less than 100 meters per second.
### Part (b)
Power of the Test:
We want to calculate the power of the test under the condition that the true mean speed is 95 meters per second.
1. True mean ([tex]\( \mu_1 \)[/tex]): 95
2. Calculate z-value for [tex]\( \mu_1 \)[/tex]:
The power of the test is given by the area to the right of the critical point for the alternative distribution.
[tex]\[
z_{\beta} = \frac{\mu_1 - \mu_0}{\sigma/\sqrt{n}} - z_{\text{critical}}
\][/tex]
3. Power Calculation:
Using the given information, the power of the test is 0.999999971.
This means that the test is almost certain to correctly reject the null hypothesis if the true mean speed is 95 meters per second.
### Part (c)
Sample Size for Desired Power:
We need to determine the sample size required to detect a mean speed as low as 93 meters per second with a power of 0.85.
1. True mean ([tex]\( \mu_1 \)[/tex]): 93
2. Desired power: 0.85
3. Calculate the required sample size:
Using the information provided, after performing the necessary statistical calculations, the required sample size is 1.
This implies that even a small sample size is sufficient to achieve a power of at least 0.85 when detecting a mean speed as low as 93 meters per second.
This completes the detailed breakdown of the solution for the given problem.
