https://file.aiquickdraw.com/mathbot/file/1726197800512d4j0iw6y.jpg

This question is asking you to calculate the test statistic for a hypothesis test concerning cleaning costs.

### Hypothesis Details:
- The null hypothesis: \( H_0: \mu = 49 \) (meaning the average cleaning cost is 49 thousand dollars).
- The significance level is \( \alpha = 0.20 \).
- You are provided with a sample dataset of cleaning costs.

### Steps to calculate the test statistic:
1. **Calculate the sample mean** \( \bar{x} \).
2. **Calculate the sample standard deviation** \( s \).
3. **Calculate the test statistic** using the formula for a one-sample t-test:
   \[
   t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
   \]
   where:
   - \( \bar{x} \) is the sample mean,
   - \( \mu_0 = 49 \) is the hypothesized population mean,
   - \( s \) is the sample standard deviation,
   - \( n \) is the sample size.

I'll calculate this now using the data provided in the image.

Let me process the numbers and compute the necessary values.The calculations are as follows:

1. **Sample mean**: \( \bar{x} = 42.22 \)
2. **Sample standard deviation**: \( s = 12.84 \)
3. **Test statistic**: \( t = -2.79 \)

This test statistic can now be compared to the critical value from a t-distribution with \( n - 1 = 27 \) degrees of freedom to determine if you should reject the null hypothesis \( H_0: \mu = 49 \) at \( \alpha = 0.20 \).

Would you like to proceed with comparing this to the critical value, or do you have any questions?

### Follow-up Questions:
1. What is the critical t-value for a significance level of \( \alpha = 0.20 \) and 27 degrees of freedom?
2. What does the negative test statistic imply in this context?
3. How do you interpret the results of the hypothesis test based on the test statistic?
4. What is the p-value corresponding to this test statistic?
5. How does increasing or decreasing the sample size affect the test statistic and its accuracy?

### Tip:
Always ensure you correctly identify the null and alternative hypotheses before conducting a test to avoid errors in interpretation.

A colleague wishes to conduct an examination of new cleaning procedures to reduce building cleaning costs. To determine if this is a promising practice, he agrees to collect data and evaluate it using α = 0.20. The null hypothesis is H0: μ = 49. Given the sample data provided (of costs in thousands), what is the test statistic for this calculation?

Learn how to calculate the test statistic for a hypothesis test on building cleaning costs with a null hypothesis of H0: μ = 49, using sample data and α = 0.20. Includes formulas for sample mean, standard deviation, and t-statistic.

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