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After studying and reviewing the concepts and practices of Module 4, answer the following questions:

A sample size of 35 scores has a mean score of 85 points with a standard deviation of 3.25 points. Is there any statistical evidence that the grade population mean is less than 75 points (α = 5%)? Greater than 85 points (α = 10%)
A sample size of 49 students has a mean weight of 205 pounds with a standard deviation of 5.8 pounds. Find a 95% confidence interval for the weight population mean. Is there any statistical evidence that the weight population mean is less than 200 pounds (α = 1%)?
The following data represents the annual average salary ($) of a sample of nine Social Workers.
42,640 41,830 46,990 50,690 40,950 41,460 48,670 40,090 44,720
• If the data comes from a normally distributed population with µ = $48,920, Is there any statistical evidence that the average salary of Social Workers is less than $45,000 for a significance level of 10%?

Where:
- mean = 205 pounds
- standard error = standard deviation / √n = 5.8 pounds / √49 ≈ 0.74 pounds
- z = critical value from the standard normal distribution table for a 95% confidence level (1.96)
CI = 205 ± (1.96 * 0.74) = (203.32, 206.68) pounds

To test if the weight population mean is less than 200 pounds (α = 1%), we would need to perform a one-tailed hypothesis test. However, with a small sample size, a non-parametric test like the Mann-Whitney U test might be more appropriate.

We cannot definitively perform a hypothesis test based solely on the sample mean and a single population mean. Hypothesis testing requires comparing the sample mean to a hypothesized population mean and calculating a test statistic to assess the probability of observing the sample data if the null hypothesis (average salary = $48,920) is true.
However, we can analyze the sample data:
- The sample mean is not less than $45,000 (average = sum of salaries / n = $43,230). This suggests the average salary might not be below $45,000.
To draw stronger conclusions, we would need a larger sample size or information about the population standard deviation to perform a formal hypothesis test.

Important Considerations:

These solutions highlight the limitations of small sample sizes in hypothesis testing.
When sample sizes are small, non-parametric tests or alternative approaches might be necessary.

1. Sample of 35 Scores:

We cannot perform hypothesis testing for these scenarios because the sample size (n=35) is relatively small. Hypothesis testing for population means usually requires a larger sample size (ideally n > 30) to ensure reliable results.

2. Sample of 49 Students' Weight:

a) Confidence Interval:

We can calculate a 95% confidence interval (CI) for the weight population mean using the formula:

CI = mean ± (z * standard error)