Are nursing salaries in Tampa, Florida, lower than those in...

Are nursing salaries in Tampa, Florida, lower than those in Dallas, Texas? As reported by the Tampa Tribune, salary data show staff nurses in Tampa earn less than staff nurses in Dallas. Suppose that in a follow-up study of 40 staff nurses in Tampa and 50 staff nurses in Dallas you obtain the following results. a. Formulate hypothesis so that, if the null hypothesis is rejected, we can conclude that salaries for staff nurses in Tampa are significantly lower than for those in Dallas. Use $\alpha=.05$. b. What is the value of the test statistic? c. What is the p-value? d. What is your conclusion?

Are nursing salaries in Tampa, Florida, lower than those in Dallas, Texas? As reported by the Tampa Tribune, salary data show staff nurses in Tampa earn less than staff nurses in Dallas. Suppose that in a follow-up study of 40 staff nurses in Tampa and 50 staff nurses in Dallas you obtain the following results.
a. Formulate hypothesis so that, if the null hypothesis is rejected, we can conclude that salaries for staff nurses in Tampa are significantly lower than for those in Dallas. Use $\alpha=.05$.
b. What is the value of the test statistic?
c. What is the p-value?
d. What is your conclusion?

Key Concepts

Conclusion of Hypothesis Testing

This concept involves interpreting the test results by comparing the p-value to the significance level. If the p-value is less than ?, the null hypothesis is rejected in favor of the alternative hypothesis, concluding that there is statistically significant evidence of a difference (in this case, that Tampa nurses earn lower salaries than Dallas nurses). If the p-value is greater than ?, the evidence is insufficient to reject the null hypothesis.

p-value

The p-value is the probability of observing the test statistic or one more extreme under the null hypothesis. It helps to measure the strength of evidence against the null hypothesis. A small p-value (less than the significance level) indicates strong evidence for rejecting the null hypothesis, leading to the conclusion that there is a statistically significant difference between the group means.

Null and Alternative Hypotheses

This concept involves stating the null hypothesis (H?) as the assumption that there is no difference or the status quo (in this case, that the mean salary for Tampa nurses is greater than or equal to that of Dallas nurses) and the alternative hypothesis (H?) proposing that the mean salary for Tampa nurses is lower than that for Dallas nurses. Formulating the correct hypotheses is crucial because a rejection of the null hypothesis results in a conclusion in support of the alternative hypothesis.

Significance Level

The significance level, denoted by ? (alpha), is a threshold for determining whether the test statistic falls in the rejection region. In the context of hypothesis testing, choosing ? = 0.05 means there is a 5% chance of committing a Type I error (incorrectly rejecting a true null hypothesis). It defines how strong the evidence must be to reject the null hypothesis.

Test Statistic and its Calculation

In comparative salary studies, especially when dealing with independent samples, the test statistic (like a t-statistic in the case of unknown population variances) is computed using the difference between sample means, pooled variances (or separate variances if conditions apply), and the sample sizes. The test statistic quantifies the difference between the groups in terms of standard error, allowing researchers to assess the standardized difference between the groups.

Transcript

00:01 So the key point that we need in order to figure out what the hypotheses are is the following statement.

00:06 If the null hypothesis is rejected, we can conclude that salaries for staff nurses in tampa bay are significantly lower than those in dallas.

00:14 So what that means is that if the alternative hypothesis, if the null hypothesis is rejected, that means the alternative hypothesis is what we go with.

00:23 And what this says is the mean salary for nurses in tampa is significantly lower than.

00:30 The mean salary for those in dallas.

00:32 So we can write our alternative hypothesis like this, or we can write it as such.

00:41 The difference between the two is less than zero.

00:48 And that means our null hypothesis is going to be that the mean salary of workers in tampa is greater than or equal to those of dallas, or that's the mean difference.

01:06 Is greater than or equal to zero.

01:10 So this is the answer to part a.

01:13 Next in part b we need to compute a test statistic and because we are given sample standard deviations instead of population standard deviations we need to compute a t test statistic and the formula for a t test statistic is as follows.

01:29 The first sample mean minus the second sample mean over square root of the first sample standard deviations squared over the first sample size plus the second standard deviation squared over the second sample size, which in this situation is equal to 56 ,100 minus 59 ,400 over 6 ,000 squared over 40 plus 7 ,000 squared over 50, which is approximately equal to negative 2 .407.

02:18 So this is our t test statistic.

02:22 Next we are going to find the degrees of freedom.

02:25 The degrees of freedom formula is complicated, so bear with me.

02:29 It is equal to first standard deviation squared over the first sample size plus the second standard deviation squared over the second sample size and the sum of those two squared over the first sample standard deviation...

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