Let's go through each part step by step:
### Step-by-Step Solution
1. Given Data:
- Variance: [tex]\(0.0037364\)[/tex]
- Number of observations ([tex]\(n\)[/tex]): 500
- Hypothesized mean ([tex]\(\mu_0\)[/tex]): 2
- t-statistic value ([tex]\(t_{\text{stat}}\)[/tex]): 1.0733641
- p-value for one-tail ([tex]\(p_{\text{one-tail}}\)[/tex]): 0.1418133
- Significance level ([tex]\(\alpha\)[/tex]): 0.05
2. Hypothesis Testing:
We are conducting a hypothesis test to determine if there is sufficient evidence to reject the null hypothesis ([tex]\(H_0\)[/tex]) which states that the population mean is equal to the hypothesized mean.
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = \mu_0\)[/tex]
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): [tex]\(\mu \neq \mu_0\)[/tex]
Since we have the p-value for a one-tailed test which is [tex]\(0.1418133\)[/tex], we need to determine whether this p-value is less than our significance level ([tex]\(\alpha\)[/tex]) of 0.05.
3. Decision Rule:
- If [tex]\(p_{\text{one-tail}} < \alpha\)[/tex], we reject the null hypothesis [tex]\(H_0\)[/tex].
- If [tex]\(p_{\text{one-tail}} \geq \alpha\)[/tex], we fail to reject the null hypothesis [tex]\(H_0\)[/tex].
4. Compare p-value and significance level:
- Given [tex]\(p_{\text{one-tail}} = 0.1418133\)[/tex]
- Given significance level [tex]\(\alpha = 0.05\)[/tex]
We observe that [tex]\(p_{\text{one-tail}} = 0.1418133 \geq \alpha = 0.05\)[/tex].
5. Conclusion:
- Since the p-value of 0.1418133 is greater than the significance level of 0.05, we fail to reject the null hypothesis.
6. Interpretation:
- There is not enough evidence to conclude that the mean weight of the eggs is significantly different from the hypothesized mean of 2. In other words, the data does not provide sufficient evidence to suggest that the mean weight of the eggs is not 2 at the 5% significance level.
### Final Statement:
Since the [tex]\(p\)[/tex]-value is 0.1418133, which is greater than the significance level of 0.05, there is insufficient evidence to reject the null hypothesis. Therefore, there is not enough evidence to suggest that the mean weight of the eggs is different from 2.
