Question

the director of state agency believes that the average starting salary for clerical employees in the state is less than $30,000 per year. to test her hypothesis, she has collected a simple random sample of 100 starting clerical salaries from across the state and found that the sample mean is $29,750. a. state the appropriate null and alternative hypotheses. b. assuming the population standard deviation is known to be $2,500 and the significance level for the test is to be 0.05, what is the critical value (stated in dollars)? c. referring to your answer in part b, what conclusion should be reached with respect to the null hypothesis? d. referring to your answer in part c, which of the two statistical errors might have been made in this case? explain.

Answer 1

The null hypothesis states that the average starting clerical salary is $30,000, while the alternative hypothesis states it is less. The critical value is $25,887.50, and since the sample mean is higher, we do not reject the null hypothesis. There's a possibility of a Type II error if the real average salary is actually less than $30,000.

Step-by-step explanation:

a. Null and Alternative Hypotheses:

Null Hypothesis (H0): The average starting salary for clerical employees in the state is $30,000 per year (H0: μ = $30,000).

Alternative Hypothesis (H1): The average starting salary for clerical employees in the state is less than $30,000 per year (H1: μ < $30,000).

b. Critical Value:

To calculate the critical value for a one-tailed z-test, we can look up the z-score corresponding to a 0.05 significance level in a standard normal distribution table since the population standard deviation is known. For a one-tailed test at α = 0.05, the critical z-score is approximately -1.645. To convert this to a dollar amount, we use the formula:

Critical value = Mean + (z-score * population standard deviation)

Critical value (in dollars) = $30,000 + (-1.645 * $2,500)

Critical value (in dollars) = $30,000 - $4,112.50 = $25,887.50

c. Conclusion:

Since the sample mean of $29,750 is higher than the critical value of $25,887.50, we do not reject the null hypothesis at the 0.05 significance level.

d. Possible Statistical Errors:

If we do not reject the null, then we may be making a Type II error if in reality, the true mean is actually less than $30,000. A Type II error occurs when we fail to reject a false null hypothesis.