To calculate the test statistic for this hypothesis test, we need to follow these steps:

Given Data and Hypotheses:

1. Sample Data: The weights of garbage discarded by 25 households.
2. Null Hypothesis ($H_0$): $\mu = 38$ lbs (the average weekly household garbage weight is 38 lbs).
3. Alternative Hypothesis ($H_1$): $\mu < 38$ lbs (the average weekly household garbage weight is less than 38 lbs).
4. Significance Level: $\alpha = 0.01$.

Steps to Compute the Test Statistic

1. Calculate the Sample Mean ($\bar{x}$):
   • Add up all the weights and divide by the number of data points (25).
2. Calculate the Sample Standard Deviation ($s$):
   • Use the formula for sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$
3. Calculate the Test Statistic (t-value):
   • Use the formula for the t-test statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$
   • Here, $\mu = 38$ lbs, $n = 25$, $\bar{x}$ is the sample mean, and $s$ is the sample standard deviation.

Calculation:

Let me calculate these values based on the provided data.The calculations yield the following results:

• Sample Mean ($\bar{x}$): 33.4880 lbs
• Sample Standard Deviation (s): 4.3147 lbs
• Test Statistic (t-value): -5.2286

This test statistic can now be compared to the critical t-value at the 1% significance level (with $n - 1 = 24$ degrees of freedom) to determine whether to reject the null hypothesis.

Would you like further clarification on any part of the calculation, or details on how to interpret the result?

Here are 5 related questions for further exploration:

1. How do we find the critical value for a 1% significance level with 24 degrees of freedom?
2. What does the negative sign in the t-statistic indicate in hypothesis testing?
3. How is the decision rule formulated for rejecting or failing to reject the null hypothesis?
4. Can we use a z-test instead of a t-test in this scenario, and why?
5. What are Type I and Type II errors in the context of this hypothesis test?

Tip: Always check the assumptions of normality, especially for small sample sizes, when performing a t-test.