Question

a. Statewide, social workers average \(10.2\) years of experience. In a random sample, 203 social workers in greater metropolitan Shinbone average only \(8.7\) years, with a standard deviation of \(0.52\). Are social workers in Shinbone significantly less experienced? (NOTE: The wording of the research hypothesis may justify a one-tailed test of significance. For a one-tailed test, what form would the research hypothesis take, and where would the critical region begin?) b. The same sample of social workers reports an average annual salary of \(\$ 25,782\), with a standard deviation of \(\$ 622\). Is this figure significantly higher than the statewide average of \(\$ 24,509\) ? (NOTE: The wording of the research hypothesis suggests a onetailed test. What form would the research hypothesis take, and where would the critical region begin?)

Short answer

Social workers in Shinbone are significantly less experienced and have a significantly higher salary than the statewide average.

Step-by-step solution

- Determine the Research Hypothesis for Experience

The research question asks whether the social workers in Shinbone are significantly less experienced. Thus, the research hypothesis for a one-tailed test is: \( H_1: \text{µ}_{Shinbone} < 10.2 \)

- Establish the Null Hypothesis and Critical Region for Experience

The null hypothesis is: \( H_0: \text{µ}_{Shinbone} = 10.2 \) For a one-tailed test, the critical region for a significance level of \( \alpha = 0.05 \) is determined by the z-value corresponding to a probability of 0.05. From z-tables, this value is approximately -1.645.

- Calculate the Test Statistic for Experience

Using the sample data, compute the z-value using the formula: \[ z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \] Substituting the values gives: \[ z = \frac{8.7 - 10.2}{\frac{0.52}{\sqrt{203}}} = \frac{-1.5}{\frac{0.52}{14.247}} = \frac{-1.5}{0.0365} \approx -41.10 \]

- Compare the Test Statistic to the Critical Value

Since \( z = -41.10 \) is much less than -1.645, reject the null hypothesis. This indicates that social workers in Shinbone have significantly less experience than the statewide average.

- Determine the Research Hypothesis for Salary

The research question asks whether the social workers in Shinbone have a significantly higher salary. Thus, the research hypothesis for a one-tailed test is: \( H_1: \text{µ}_{Shinbone} > 24,509 \)

- Establish the Null Hypothesis and Critical Region for Salary

The null hypothesis is: \( H_0: \text{µ}_{Shinbone} = 24,509 \) For a one-tailed test, the critical region for a significance level of \( \alpha = 0.05 \) is determined by the z-value corresponding to a probability of 0.95. From z-tables, this value is approximately 1.645.

- Calculate the Test Statistic for Salary

Using the sample data, compute the z-value using the formula: \[ z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \] Substituting the values gives: \[ z = \frac{25,782 - 24,509}{\frac{622}{\sqrt{203}}} = \frac{1,273}{\frac{622}{14.247}} = \frac{1,273}{43.676} \approx 29.14 \]

- Compare the Test Statistic to the Critical Value

Since \( z = 29.14 \) is much greater than 1.645, reject the null hypothesis. This indicates that social workers in Shinbone have a significantly higher salary than the statewide average.

Key concepts

One-Tailed Test

A one-tailed test in hypothesis testing is used to determine if a sample mean is significantly greater or less than the population mean in a specific direction. Unlike a two-tailed test, which checks for any significant difference in either direction, a one-tailed test is directional.
For example, in the provided exercise, we have two situations:

• Experience: We are checking if the social workers in Shinbone have significantly less experience than the statewide average. Here, the hypothesis test is one-tailed going in the negative direction.
• Salary: We are checking if the social workers in Shinbone have significantly higher salaries than the statewide average. This is a one-tailed test in the positive direction.

It's important to choose a one-tailed test when you have a clear expectation about the direction of the effect.

Z-Value

The z-value, or z-score, is a statistical measurement that describes the position of a raw score in terms of its distance from the mean, measured in standard deviations. It's a critical part of hypothesis testing because it helps to determine how far away our sample statistic is from the population parameter.
The formula for the z-value is:
\[ z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \] where:

• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean
• \( s \) is the standard deviation of the sample
• \( n \) is the sample size

In our example, for the experience of social workers in Shinbone, we calculate a z-value of approximately -41.10. For salary, we calculate a z-value of approximately 29.14. These z-values are compared against critical values from standard z tables to decide whether to reject the null hypothesis.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement that there is no effect or no difference, and it serves as the default or starting assumption in hypothesis testing. It's what we attempt to test against.
In the context of our exercise, we have two null hypotheses:

• For experience: \(H_0: µ_{Shinbone} = 10.2\). This means that we assume the average years of experience among social workers in Shinbone is equal to the statewide average of 10.2 years until proven otherwise.
• For salary: \(H_0: µ_{Shinbone} = 24,509\). Here, we assume that the average annual salary among social workers in Shinbone is equal to the statewide average of $24,509 until proven otherwise.

Rejecting the null hypothesis would suggest that there is a significant difference in the means.

Research Hypothesis

The research hypothesis (\(H_1\)) is the statement we are trying to find evidence to support. It reflects our specific claim or prediction about the population parameter.
In our example, the research hypotheses are:

• For experience: \(H_1: µ_{Shinbone} < 10.2\). This suggests that social workers in Shinbone have significantly less average experience than the statewide average.
• For salary: \(H_1: µ_{Shinbone} > 24,509\). This proposes that social workers in Shinbone have a significantly higher average salary than the statewide average.

The research hypothesis directly opposes the null hypothesis, and when we reject the null hypothesis based on our test statistics, we support the research hypothesis, indicating a significant finding in the stated direction.