Alright, let's carefully analyze the given data and draw a conclusion step-by-step:
### Step-by-Step Analysis
1. Understanding the Hypothesis:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean shipping time is 2 days, i.e., [tex]\(\mu = 2\)[/tex].
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): The mean shipping time is greater than 2 days, i.e., [tex]\(\mu > 2\)[/tex].
2. Given Data:
- Sample Mean ([tex]\(\bar{x}\)[/tex]) = 2.05602454
- Hypothesized Mean ([tex]\(\mu_0\)[/tex]) = 2
- Sample Variance (s²) = 0.48326032
- Sample Size (n) = 400
- Degrees of Freedom (df) = 399
- Significance Level ([tex]\(\alpha\)[/tex]) = 0.05
- t-Statistic (t) = 1.611824412
- P-value (one-tail) = 0.053895421
- Critical t-value (one-tail) = 1.648681534
3. Interpret the t-Statistic and Critical t-value:
- The t-statistic represents how far our sample mean is from the hypothesized mean in terms of standard errors.
- The critical t-value at the 0.05 significance level for a one-tail test is 1.648681534.
4. Compare t-statistic with Critical t-value:
- The given t-statistic (1.6118) is less than the critical t-value (1.6487). Hence, the observed sample mean does not fall in the rejection region.
5. Interpret the p-value:
- The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed under the null hypothesis.
- The given one-tailed p-value is 0.0539.
6. Decision Making:
- We compare the p-value to the significance level ([tex]\(\alpha = 0.05\)[/tex]).
- Since 0.0539 > 0.05, we do not reject the null hypothesis.
### Conclusion
Since the p-value (0.0539) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the mean shipping time is greater than 2 days from the given sample. The observed sample does not provide sufficient evidence to support the advocacy group’s claim that the shipping times are longer than the promised 2 days.
