Answer :
### Solution
Let's tackle the problem step by step:
---
(a) Verifying Conditions for Hypothesis Testing
To conduct a hypothesis test, especially a t-test, we check the following conditions:
1. Random Sampling: The sample of students should be randomly selected to ensure it is representative of the population.
2. Independence: Each GPA measurement should be independent of the others. Given that the sample size (22) is much less than 10% of the population size (5000), the independence condition is satisfied.
3. Normality: For small sample sizes (usually less than 30), we assume the sample data comes from a normally distributed population or the sample data itself should be approximately normal.
Given the sample size and no reason to suspect non-normality of GPAs, we can proceed with the t-test.
---
(b) Hypothesis Testing at the 5% Significance Level
Step-by-Step Process:
1. State the Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The average GPA is greater than or equal to 3.15.
[tex]\[ H_0: \mu \geq 3.15 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The average GPA is less than 3.15.
[tex]\[ H_1: \mu < 3.15 \][/tex]
2. Collect Sample Data:
[tex]\[ \{2, 2.3, 2.11, 2.78, 4, 2.7, 2.6, 3.7, 3.75, 2.82, 3.2, 3, 2.85, 3, 3.13, 3, 2.8, 3.04, 2.8, 2.5, 3.5, 2.28\} \][/tex]
3. Calculate Sample Mean and Standard Deviation:
- Sample Mean, [tex]\( \bar{x} \)[/tex]: [tex]\( 2.9027 \)[/tex]
- Sample Standard Deviation, [tex]\( s \)[/tex]: [tex]\( 0.5181 \)[/tex]
- Sample Size, [tex]\( n \)[/tex]: [tex]\( 22 \)[/tex]
4. Determine the T-Statistic:
The t-statistic formula is given by:
[tex]\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \][/tex]
Plugging in the values:
[tex]\[ t = \frac{2.9027 - 3.15}{\frac{0.5181}{\sqrt{22}}} \approx -2.239 \][/tex]
5. Calculate the p-value:
Use a t-distribution with [tex]\( n-1 = 21 \)[/tex] degrees of freedom to find the p-value corresponding to the t-statistic of -2.239.
[tex]\[ \text{p-value} = 0.0181 \][/tex]
6. Compare the p-value with the Significance Level ([tex]\( \alpha \)[/tex]):
[tex]\[ \alpha = 0.05 \][/tex]
Since the p-value (0.0181) is less than 0.05, we reject the null hypothesis.
7. Conclusion:
At the 5% level of significance, we reject the null hypothesis. This means there is enough evidence to support the CampusReel.com claim that the average GPA at Jackson College is lower than 3.15.
---
In summary:
- Sample Mean: [tex]\( 2.9027 \)[/tex]
- Sample Standard Deviation: [tex]\( 0.5181 \)[/tex]
- t-Statistic: [tex]\( -2.239 \)[/tex]
- p-Value: [tex]\( 0.0181 \)[/tex]
- Conclusion: Reject the null hypothesis. There is enough evidence to support the claim that the average GPA is lower than 3.15.
Let's tackle the problem step by step:
---
(a) Verifying Conditions for Hypothesis Testing
To conduct a hypothesis test, especially a t-test, we check the following conditions:
1. Random Sampling: The sample of students should be randomly selected to ensure it is representative of the population.
2. Independence: Each GPA measurement should be independent of the others. Given that the sample size (22) is much less than 10% of the population size (5000), the independence condition is satisfied.
3. Normality: For small sample sizes (usually less than 30), we assume the sample data comes from a normally distributed population or the sample data itself should be approximately normal.
Given the sample size and no reason to suspect non-normality of GPAs, we can proceed with the t-test.
---
(b) Hypothesis Testing at the 5% Significance Level
Step-by-Step Process:
1. State the Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The average GPA is greater than or equal to 3.15.
[tex]\[ H_0: \mu \geq 3.15 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The average GPA is less than 3.15.
[tex]\[ H_1: \mu < 3.15 \][/tex]
2. Collect Sample Data:
[tex]\[ \{2, 2.3, 2.11, 2.78, 4, 2.7, 2.6, 3.7, 3.75, 2.82, 3.2, 3, 2.85, 3, 3.13, 3, 2.8, 3.04, 2.8, 2.5, 3.5, 2.28\} \][/tex]
3. Calculate Sample Mean and Standard Deviation:
- Sample Mean, [tex]\( \bar{x} \)[/tex]: [tex]\( 2.9027 \)[/tex]
- Sample Standard Deviation, [tex]\( s \)[/tex]: [tex]\( 0.5181 \)[/tex]
- Sample Size, [tex]\( n \)[/tex]: [tex]\( 22 \)[/tex]
4. Determine the T-Statistic:
The t-statistic formula is given by:
[tex]\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \][/tex]
Plugging in the values:
[tex]\[ t = \frac{2.9027 - 3.15}{\frac{0.5181}{\sqrt{22}}} \approx -2.239 \][/tex]
5. Calculate the p-value:
Use a t-distribution with [tex]\( n-1 = 21 \)[/tex] degrees of freedom to find the p-value corresponding to the t-statistic of -2.239.
[tex]\[ \text{p-value} = 0.0181 \][/tex]
6. Compare the p-value with the Significance Level ([tex]\( \alpha \)[/tex]):
[tex]\[ \alpha = 0.05 \][/tex]
Since the p-value (0.0181) is less than 0.05, we reject the null hypothesis.
7. Conclusion:
At the 5% level of significance, we reject the null hypothesis. This means there is enough evidence to support the CampusReel.com claim that the average GPA at Jackson College is lower than 3.15.
---
In summary:
- Sample Mean: [tex]\( 2.9027 \)[/tex]
- Sample Standard Deviation: [tex]\( 0.5181 \)[/tex]
- t-Statistic: [tex]\( -2.239 \)[/tex]
- p-Value: [tex]\( 0.0181 \)[/tex]
- Conclusion: Reject the null hypothesis. There is enough evidence to support the claim that the average GPA is lower than 3.15.
