/*! 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 14 Critical Job Skills In August \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 14

Critical Job Skills In August \(2003,56 \%\) of employed adults in the United States reported that basic mathematical skills were critical or very important to their job. The supervisor of the job placement office at a 4 -year college thinks this percentage has increased due to increased use of technology in the workplace. He takes a random sample of 480 employed adults and finds that 297 of them feel that basic mathematical skills are critical or very important to their job. Has the percentage of employed adults who feel basic mathematical skills are critical or very important to their job increased? Use the \(\alpha=0.05\) level of significance.

Short Answer

Expert verified
Reject the null hypothesis. The percentage of employed adults who feel basic mathematical skills are critical has increased.

Step by step solution

01

Define Hypotheses

State the null hypothesis (\text{H}_0) and the alternative hypothesis (\text{H}_a).\[ \text{H}_0: p = 0.56 \]\[ \text{H}_a: p > 0.56 \]We are testing if the proportion has increased, so the alternative hypothesis is that the proportion is greater than 0.56.
02

Sample Proportion

Calculate the sample proportion (\text{\text{p-hat}}) of employed adults who feel that mathematical skills are important.\[ \text{p-hat} = \frac{297}{480} \ = 0.61875 \]
03

Standard Error Calculation

Calculate the standard error (SE) for the sample proportion.\[ SE = \sqrt{\frac{p(1-p)}{n}} \ = \sqrt{\frac{0.56(1-0.56)}{480}} \ = 0.0227 \]
04

Test Statistic Calculation

Compute the test statistic (z).\[ z = \frac{\text{p-hat} - p}{SE} \ = \frac{0.61875 - 0.56}{0.0227} \ \ = 2.59 \]
05

Determine Critical Value and P-value

For a right-tailed test at \(\text{\text{伪=0.05}}\), find the critical value and the p-value.The critical value z for \(\text{\text{伪=0.05}}\text{\text{}}\) is 1.645.Using a z-table or calculator, find the p-value for z = 2.59, which is approximately 0.0048.
06

Conclusion

Compare the calculated p-value to \(\text{伪}\):Since the p-value (=0.0048) is less than \(\text{伪} = 0.05\), reject the null hypothesis. There is sufficient evidence to conclude that the percentage of employed adults in the U.S. who feel basic mathematical skills are critical has increased.

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.

Null Hypothesis
In hypothesis testing, the null hypothesis (\text{H}_0) is a statement of no effect or no difference. It serves as a starting point for statistical analysis. For this problem, the null hypothesis is that the proportion of employed adults who believe mathematical skills are critical has not changed since 2003. Mathematically, this is expressed as:\[ \text{H}_0: p = 0.56 \]Exploring the null hypothesis is crucial because it provides a benchmark against which we can measure any observed differences.
Alternative Hypothesis
The alternative hypothesis (\text{H}_a) suggests that there has been a change or effect. It is what the researcher is trying to prove. In this case, we want to determine if the percentage has increased. Thus, the alternative hypothesis is:\[ \text{H}_a: p > 0.56 \]Confirming the alternative hypothesis would indicate an increase in the perception that mathematical skills are critical, reflecting perhaps more reliance on technology in the workplace.
Sample Proportion
The sample proportion (\text{p-hat}) measures the fraction of our sample that meets a certain condition. It gives us a practical, empirical basis for our testing. In this study, the sample proportion of adults who feel mathematical skills are critical can be calculated as follows:\[\text{p-hat} = \frac{297}{480} = 0.61875\]This value is higher than the hypothesized 56%, which suggests an increase. But to confirm this, further analysis is required.
Standard Error
The standard error (SE) tells us how much the sample proportion (\text{p-hat}) is expected to fluctuate from the true population proportion (p). It is vital in estimating the accuracy of our sample statistic. The formula for standard error in this exercise is:\[SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.56(1-0.56)}{480}} = 0.0227\]The smaller the SE, the more representative our sample is of the population.
Test Statistic
The test statistic (z) quantifies the degree to which the sample proportion deviates from the null hypothesis value, adjusted for the standard error. It is calculated as:\[ z = \frac{\text{p-hat} - p}{SE} = \frac{0.61875 - 0.56}{0.0227} = 2.59 \]A higher absolute value of the test statistic usually indicates stronger evidence against the null hypothesis.
P-value
The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. For the computed z-value of 2.59, the p-value is approximately 0.0048. Since this value is less than our significance level (\(\alpha = 0.05\)), it suggests that the observed effect is statistically significant, leading us to reject the null hypothesis.
Critical Value
The critical value is a threshold set by the significance level (\(\alpha\)) that determines whether the null hypothesis should be rejected. For a right-tailed test at \(\alpha = 0.05\), the critical value is 1.645. If the test statistic exceeds this critical value, it falls in the rejection region. Since our test statistic (2.59) is greater than 1.645, we reject the null hypothesis, indicating significant evidence that the percentage of adults considering mathematical skills as critical has increased.

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

Test the hypothesis, using (a) the classical approach and then (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{aligned}&H_{0}: p=0.4 \text { versus } H_{1}: p \neq 0.4\\\&n=1000 ; x=420 ; \alpha=0.01\end{aligned}$$

To test \(H_{0}: \mu=20\) versus \(H_{1}: \mu<20,\) a simple random sample of size \(n=18\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=18.3\) and \(s=4.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

Simulation Simulate drawing 50 simple random samples of size \(n=20\) from a population that is normally distributed with mean 80 and standard deviation \(7 .\) (a) Test the null hypothesis \(H_{0}: \mu=80\) versus the alternative hypothesis \(H_{1}: \mu \neq 80\) (b) Suppose we were testing this hypothesis at the \(\alpha=0.1\) level of significance. How many of the 50 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) Describe how we know that a rejection of the null hypothesis results in making a Type I error in this situation.

To test \(H_{0}: \mu=80\) versus \(H_{1}: \mu<80,\) a simple random sample of size \(n=22\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=76.9\) and \(s=8.5,\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.02\) level of significance, determine the critical value. (c) Draw a \(t\) -distribution that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why?

(a) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, and (c) explain what it would mean to make a Type II error. Federal law requires that a jar of peanut butter that is labeled as containing 32 ounces must contain at least 32 ounces. A consumer advocate feels that a certain peanut butter manufacturer is shorting customers by under filling the jars so that the mean content is less than the 32 ounces stated on the label.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

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.