Question

Perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Teachers' Salaries New York and Massachusetts lead the list of average teacher's salaries. The New York average is \(\$ 76,409\) while teachers in Massachusetts make an average annual salary of \(\$ 73,195 .\) Random samples of 45 teachers from each state yielded the following. $$ \begin{array}{lrr} & \text { Massachusetts } & \text { New York } \\ \hline \text { Sample means } & \$ 73,195 & \$ 76,409 \\ \text { Population standard deviation } & 8,200 & 7,800 \end{array} $$ At \(\alpha=0.10\), is there a difference in means of the salaries?

Short answer

There is a significant difference in salaries.

Step-by-step solution

State the Hypotheses and Identify the Claim

To determine whether there is a statistically significant difference in the means of the salaries between teachers in Massachusetts and New York, we start by stating the null and alternative hypotheses. Let \( \mu_1 \) be the mean salary for Massachusetts teachers and \( \mu_2 \) be the mean salary for New York teachers.- Null Hypothesis (\( H_0 \)): \( \mu_1 = \mu_2 \) (There is no difference in the means.)- Alternative Hypothesis (\( H_1 \)): \( \mu_1 eq \mu_2 \) (There is a difference in the means.)The alternative hypothesis represents the claim that we are testing.

Find the Critical Value(s)

Given \( \alpha = 0.10 \) for a two-tailed test, we need to find the critical \( z \)-values. Since it's a two-tailed test, we divide \( \alpha \) by 2 to get \( \alpha/2 = 0.05 \) on each tail. Using a standard normal distribution table or calculator, we find the critical \( z \)-values: \[ z = \pm 1.645 \]

Compute the Test Value

Next, we compute the test statistic, using the formula for the difference between two means:\[ z = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]where:- \( \bar{X}_1 = 73195 \)- \( \bar{X}_2 = 76409 \)- \( \sigma_1 = 8200 \)- \( \sigma_2 = 7800 \)- \( n_1 = n_2 = 45 \)Substitute the values:\[ z = \frac{(73195 - 76409)}{\sqrt{\frac{8200^2}{45} + \frac{7800^2}{45}}} \] \[ z = \frac{-3214}{\sqrt{1484444.44 + 1352000}} \] \[ z = \frac{-3214}{\sqrt{2836444.44}} \] \[ z = \frac{-3214}{1683.05} \] \[ z \approx -1.91 \]

Make the Decision

The computed test statistic \( z = -1.91 \) is compared against the critical \( z \)-values \( \pm 1.645 \).Since \( -1.91 \) does not fall within the acceptance region (i.e., it is less than \(-1.645\)), we reject the null hypothesis.

Summarize the Results

At the 0.10 significance level, there is enough statistical evidence to reject the null hypothesis. Thus, we conclude that there is a significant difference in the means of teacher salaries between Massachusetts and New York.

Key concepts

Critical Value

In traditional hypothesis testing, the critical value is a key component used to decide whether to reject a null hypothesis. It acts as a threshold that the test statistic must exceed for us to conclude that there is a significant effect.
Critical values are determined by the significance level (alpha, \( \alpha \)), which represents the probability of rejecting a true null hypothesis. Common values for \( \alpha \) are 0.05 and 0.10, reflecting a 5% and 10% risk, respectively, of incorrectly rejecting the null hypothesis.
In a two-tailed test, the significance level is split between the two tails of the distribution, where each tail accounts for half of \( \alpha \).

• For the given problem with \( \alpha = 0.10 \), each tail would have \( \alpha/2 = 0.05 \).
• Using a standard normal distribution table, the critical values corresponding to \( \alpha = 0.10 \) are \( \pm 1.645 \).

The critical values define the points beyond which we consider the observed effect not to be a result of random chance.

Null Hypothesis

The null hypothesis, symbolized as \( H_0 \), is a statement positing that there is no effect or no difference, and it serves as the baseline assumption in hypothesis testing. It is essentially a position of 'no change'.
In our exercise, the null hypothesis is \( H_0: \mu_1 = \mu_2 \), indicating that there is no difference in average teacher salaries between Massachusetts and New York. Null hypotheses are essential because they provide a starting point for statistical testing and are what we attempt to disprove or reject through our analysis.
When we conduct a hypothesis test, we actually test the null hypothesis. If the statistical evidence is strong enough to reject \( H_0 \), we may conclude that there is support for the alternative hypothesis \( H_1 \), which in this case suggests that \( \mu_1 eq \mu_2 \).

Test Statistic

A test statistic is a numerical value calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis. The formula for the test statistic depends on the statistical test being used and the type of data.
For comparing two means, as in our exercise, we use the formula for the \( z \)-test statistic:
\[ z = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]

• \( \bar{X}_1 \) and \( \bar{X}_2 \) are the sample means for Massachusetts and New York, respectively.
• \( \sigma_1 \) and \( \sigma_2 \) are the population standard deviations for each state.
• \( n_1 \) and \( n_2 \) are the sample sizes.

In this problem, the calculated test statistic was approximately \( z = -1.91 \). This value is then compared to the critical value to make a decision.

Two-Tailed Test

A two-tailed test is a hypothesis test that considers both directions of effect: above or below the hypothesized parameter. This test is used when we are interested in determining whether there is a difference in either direction, not just a specific one.
For example, if you're assuming that two means are equal, as with the teacher salaries from Massachusetts and New York, a two-tailed test would check if the means are either greater or less than each other.

• In a two-tailed test, the significance level \( \alpha \) is divided equally between the two tails of the probability distribution.
• This means that each tail has a critical region of \( \alpha/2 \), which guards against extreme values on both ends.

In our example with a significance level of \( \alpha = 0.10 \), we allocate 0.05 to each tail. If the test statistic falls into either of these critical regions (beyond \( \pm 1.645 \)), we have strong evidence against the null hypothesis, suggesting a significant difference in teacher salaries between the states.