Answer :
To solve this problem, we need to perform a two-sample t-test to compare the means of two independent samples. Here is a step-by-step approach:
### Step 1: Understand the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu_1 = \mu_2\)[/tex], meaning the means of the two samples are equal.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu_1 \neq \mu_2\)[/tex], meaning the means of the two samples are not equal.
### Step 2: Gather the Data
- Sample 1 consists of the following values:
[tex]\[
[104.3, 73.8, 71.4, 77.7, \ldots , 90.8] \quad \text{(41 values total)}
\][/tex]
- Sample 2 consists of the following values:
[tex]\[
[92.8, 59.6, 90.1, 82.6, \ldots , 68.7] \quad \text{(44 values total)}
\][/tex]
### Step 3: Perform the Two-Sample t-Test
The two-sample t-test compares the means of the two samples assuming they come from populations with equal variances and the samples are independent and normally distributed.
### Step 4: Calculate the Test Statistic
The test statistic reflects the difference between the sample means relative to the variability of the samples.
### Step 5: Determine the p-value
The p-value indicates the probability of observing the test statistic or something more extreme if the null hypothesis is true.
### Results
- Test Statistic: [tex]\(2.139\)[/tex]
- p-value: [tex]\(0.0354\)[/tex]
### Step 6: Decision
- Compare the p-value to the significance level [tex]\(\alpha = 0.005\)[/tex].
- Since [tex]\(0.0354 > 0.005\)[/tex], we fail to reject the null hypothesis.
### Conclusion
There is not enough statistical evidence at the [tex]\(0.005\)[/tex] significance level to conclude that there is a difference in the means of the two samples. The data does not show a significant difference between the group means.
### Step 1: Understand the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu_1 = \mu_2\)[/tex], meaning the means of the two samples are equal.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu_1 \neq \mu_2\)[/tex], meaning the means of the two samples are not equal.
### Step 2: Gather the Data
- Sample 1 consists of the following values:
[tex]\[
[104.3, 73.8, 71.4, 77.7, \ldots , 90.8] \quad \text{(41 values total)}
\][/tex]
- Sample 2 consists of the following values:
[tex]\[
[92.8, 59.6, 90.1, 82.6, \ldots , 68.7] \quad \text{(44 values total)}
\][/tex]
### Step 3: Perform the Two-Sample t-Test
The two-sample t-test compares the means of the two samples assuming they come from populations with equal variances and the samples are independent and normally distributed.
### Step 4: Calculate the Test Statistic
The test statistic reflects the difference between the sample means relative to the variability of the samples.
### Step 5: Determine the p-value
The p-value indicates the probability of observing the test statistic or something more extreme if the null hypothesis is true.
### Results
- Test Statistic: [tex]\(2.139\)[/tex]
- p-value: [tex]\(0.0354\)[/tex]
### Step 6: Decision
- Compare the p-value to the significance level [tex]\(\alpha = 0.005\)[/tex].
- Since [tex]\(0.0354 > 0.005\)[/tex], we fail to reject the null hypothesis.
### Conclusion
There is not enough statistical evidence at the [tex]\(0.005\)[/tex] significance level to conclude that there is a difference in the means of the two samples. The data does not show a significant difference between the group means.
