/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Mary Jo Fitzpatrick is the vice ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 6

Mary Jo Fitzpatrick is the vice president for Nursing Services at St. Luke's Memorial Hospital. Recently she noticed in the job postings for nurses that those that are unionized seem to offer higher wages. She decided to investigate and gathered the following information. $$ \begin{array}{|lccc|} \hline & & \text { Population } & \\ \text { Group } & \text { Mean Wage } & \text { Standard Deviation } & \text { Sample Size } \\ \hline \text { Union } & \$ 20.75 & \$ 2.25 & 40 \\ \text { Nonunion } & \$ 19.80 & \$ 1.90 & 45 \\ \hline \end{array} $$ Would it be reasonable for her to conclude that union nurses earn more? Use the .02 significance level. What is the \(p\) -value?

Short Answer

Expert verified
Yes, it is reasonable; the test statistic is significant at \( \alpha = 0.02 \), and the p-value supports it.

Step by step solution

01

State the Hypotheses

We need to determine whether union nurses earn more than nonunion nurses. We set up our hypotheses as follows: \( H_0: \mu_1 = \mu_2 \) (Union and nonunion nurses earn the same) \( H_a: \mu_1 > \mu_2 \) (Union nurses earn more than nonunion nurses). This is a one-tailed test.
02

Identify the Level of Significance

The significance level provided is \( \alpha = 0.02 \). This means we will reject the null hypothesis if our test statistic falls into the critical region defined by \( \alpha = 0.02 \).
03

Calculate the Test Statistic

We will use a two-sample t-test to compare the means:\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]Where:\( \bar{x}_1 = 20.75 \), \( \bar{x}_2 = 19.80 \)\( s_1 = 2.25 \), \( s_2 = 1.90 \)\( n_1 = 40 \), \( n_2 = 45 \)Plug in the values:\[ t = \frac{20.75 - 19.80}{\sqrt{\frac{2.25^2}{40} + \frac{1.90^2}{45}}} \]
04

Calculate Degrees of Freedom

Use the formula for the degrees of freedom for a two-sample t-test:\[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]Plug in the values to find \( df \).
05

Determine Critical Value and Compare the Test Statistic

With the calculated degrees of freedom, find the critical t-value at \( \alpha = 0.02 \) using a t-distribution table. Compare it with the calculated test statistic from Step 3. If the test statistic is greater than the critical value, reject the null hypothesis.
06

Calculate the p-value

Use a t-distribution table or statistical software to find the p-value associated with the calculated t statistic and the degrees of freedom. This p-value will indicate the probability of observing the computed statistic if the null hypothesis is true.
07

Conclusion

Based on the comparison in Step 5 and the p-value from Step 6, determine if \( H_0 \) should be rejected or not. If the p-value is less than \( \alpha = 0.02 \), reject \( H_0 \) and conclude that union nurses earn more.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Sample t-Test
When comparing the wages of union and non-union nurses, a two-sample t-test is an effective tool. This statistical test helps us determine if there is a significant difference between the means of two independent groups. Here, we compare the mean wage of union nurses (\( \bar{x}_1 = 20.75 \)) with that of non-union nurses (\( \bar{x}_2 = 19.80 \)).
The formula for the two-sample t-test statistic is given by:\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]- \( s_1 \) and \( s_2 \) are the standard deviations of the union and non-union groups, respectively.- \( n_1 \) and \( n_2 \) are their respective sample sizes.
This test assumes that the data from both groups have a normal distribution and that the variances are equal. This helps in asserting if the observed difference in means is statistically significant or just due to random chance.
Significance Level
In hypothesis testing, the significance level (\( \alpha \)) is crucial. It represents the probability of rejecting the null hypothesis when it is actually true, also known as the Type I error.
Mary Jo decided on a significance level of 0.02 for her test. This means she is willing to accept a 2% risk of concluding that union wages are greater when they are not.
  • A smaller significance level means a more conservative test, reducing the risk of a false positive result.
  • This choice impacts decisions guided by probability tables, where we compare our test statistic to a critical value.
A significance level of 0.02 suggests high confidence in results, implying a stringent criterion for rejecting the null hypothesis.
Degrees of Freedom
Degrees of freedom (df) are a concept important in calculating the test statistic for the two-sample t-test. They show how many values in calculating a statistic are free to vary, taking into account the constraints in the problem.
The calculation of degrees of freedom for a two-sample t-test is given by:\[df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}\]This formula helps provide an accurate critical value during the hypothesis testing process.
The degrees of freedom adjust based on the differing sample sizes and standard deviations of the two groups, influencing the shape and spread of the t-distribution used to determine critical values and p-values.
p-Value
The p-value is a measure that helps in decision-making during hypothesis testing. It tells us the probability of observing test results at least as extreme as those observed, assuming the null hypothesis is true.
Once the test statistic is calculated, the p-value is obtained by comparing it against a standard t-distribution.
  • If the p-value is less than the chosen significance level (\( \alpha = 0.02 \) in this case), then we reject the null hypothesis.
  • A smaller p-value indicates stronger evidence against the null hypothesis.
  • In this exercise, if the p-value is below 0.02, it suggests strong evidence that union nurses indeed earn more than non-union nurses.
By evaluating the p-value, decision-makers can determine the statistical validity of their alternative hypothesis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A recent study compared the time spent together by single- and dual-earner couples. According to the records kept by the wives during the study, the mean amount of time spent together watching television among the single-earner couples was 61 minutes per day, with a standard deviation of 15.5 minutes. For the dual-earner couples, the mean number of minutes spent watching television was 48.4 minutes, with a standard deviation of 18.1 minutes. At the .01 significance level, can we conclude that the single-earner couples on average spend more time watching television together? There were 15 single-earner and 12 dual-earner couples studied.

Fry Brothers Heating and Air Conditioning, Inc. employs Larry Clark and George Murnen to make service calls to repair furnaces and air conditioning units in homes. Tom Fry, the owner, would like to know whether there is a difference in the mean number of service calls they make per day. Assume the population standard deviation for Larry Clark is 1.05 calls per day and 1.23 calls per day for George Murnen. A random sample of 40 days last year showed that Larry Clark made an average of 4.77 calls per day. For a sample of 50 days George Murnen made an average of 5.02 calls per day. At the .05 significance level, is there a difference in the mean number of calls per day between the two employees? What is the \(p\) -value?

The Roper Organization conducted identical surveys 5 years apart. One question asked of women was "Are most men basically kind, gentle, and thoughtful?" The earlier survey revealed that, of the 3,000 women surveyed, 2,010 said that they were. The later revealed 1,530 of the 3,000 women surveyed thought that men were kind, gentle, and thoughtful. At the .05 level, can we conclude that women think men are less kind, gentle, and thoughtful in the later survey compared with the earlier one?

As part of a study of corporate employees, the director of human resources for PNC, Inc., wants to compare the distance traveled to work by employees at its office in downtown Cincinnati with the distance for those in downtown Pittsburgh. A sample of 35 Cincinnati employees showed they travel a mean of 370 miles per month. A sample of 40 Pittsburgh employees showed they travel a mean of 380 miles per month. The population standard deviations for the Cincinnati and Pittsburgh employees are 30 and 26 miles, respectively. At the .05 significance level, is there a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees?

A nationwide sample of influential Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur coal could be burned in coal-fired power plants. The results were: $$ \begin{array}{|lcc|} \hline & \text { Republicans } & \text { Democrats } \\ \hline \text { Number sampled } & 1,000 & 800 \\ \text { Number in favor } & 200 & 168 \end{array} $$ At the .02 level of significance, can we conclude that there is a larger proportion of Democrats in favor of lowering the standards? Determine the \(p\) -value.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.