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Chapter 13: Problem 71

The employee relations manager of a large company was concerned that raises given to employees during a recent period might not have been based strictly on objective performance criteria. A sample of \(n=20\) employees was selected, and the values of \(x\), a quantitative measure of productivity, and \(y\), the percentage salary increase, were determined for each one. A computer package was used to fit the simple linear regression model, and the resulting output gave the \(P\) -value \(=.0076\) for the model utility test. Does the percentage raise appear to be linearly related to productivity? Explain.

Short Answer

Expert verified
Yes, based on the p-value of .0076 which is less than the standard significance level (0.05), it can be concluded that the percentage salary raise appears to have a linear relationship with the employees' productivity.

Step by step solution

01

Understanding the portrayed scenario

This situation involves a test of a linear regression model. The measure of productivity (\(x\)) and the percentage salary increase (\(y\)) for each employee were the variables determined in the collected data.
02

Interpreting the P-value

The P-value obtained from the regression analysis is .0076. Recall that a general rule of thumb in interpretation is if the P-value is less than a predetermined significance level (usually 0.05), there is sufficient evidence to reject the null hypothesis, which generally postulates no relationship between the variables.
03

Conclusion of the relationship

Given that the P-value (.0076) is less than 0.05, there is sufficient evidence to reject the null hypothesis of no relationship. Hence, it can be inferred that the productivity level of an employee does appear to have a linear relationship with the percentage of their salary increase.

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Key Concepts

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

P-value interpretation
Understanding the P-value is crucial in statistical analysis, especially when determining the relevance of data in tests like simple linear regression. The P-value helps us understand the probability that the observed results would occur under the null hypothesis. It is a significant numerical tool used in hypothesis testing.

When interpreting the P-value:
  • A smaller P-value indicates stronger evidence against the null hypothesis, suggesting that there is a true effect or relationship.
  • Commonly, a P-value threshold of 0.05 is used as a benchmark. If the P-value is below this level, we typically reject the null hypothesis.
  • In this exercise, the P-value was 0.0076, which is significantly smaller than 0.05. This suggests a strong linear relationship between productivity and salary increase.
Interpreting the P-value correctly allows us to draw meaningful conclusions from data, affirming the results of our statistical test.
Null hypothesis testing
Null hypothesis testing is a fundamental concept in statistics that allows researchers to determine whether there is enough evidence to support a specific claim about a population parameter. In the context of simple linear regression, the null hypothesis generally states that there is no linear relationship between the independent variable and the dependent variable.

Key aspects of Null Hypothesis Testing:
  • The null hypothesis (often denoted as \(H_0\)) proposes there is no effect or no relationship.
  • The alternative hypothesis (denoted as \(H_a\)) suggests that there is an effect or a relationship.
  • Using statistical tests, researchers assess the likelihood of the null hypothesis being true. If the evidence against it is strong enough, they reject the null hypothesis in favor of the alternative.
In our scenario, the null hypothesis postulates no relationship between productivity and salary increase. Since the P-value is sufficiently low (0.0076), we reject the null hypothesis, concluding that there is indeed a significant linear relationship present.
Statistical significance
Statistical significance is a term used to describe when the results of a statistical test indicate a relationship or effect that is unlikely to have occurred by chance alone. It is a measurement that helps in validating whether the findings from a sample can be reasonably expected to apply to the larger population.

Understanding Statistical Significance:
  • It is linked to the P-value; if the P-value is below a predetermined threshold (commonly 0.05), results are considered statistically significant.
  • Statistical significance does not imply practical significance, meaning that while results might be statistically valid, they may not be meaningful in real-world applications.
  • In the given exercise where the P-value is 0.0076, which is below 0.05, the relationship between productivity and salary increase is statistically significant.
This means the observed correlation is strong enough for us to conclude it is unlikely due to random variation, and we can have confidence in the relationship indicated by the sample data.

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Most popular questions from this chapter

Data on \(x=\) depth of flooding and \(y=\) flood damage were given in Exercise 5.75. Summary quantities are $$ \begin{aligned} &n=13 \quad \sum x=91 \quad \sum x^{2}=819 \\ &\sum y=470 \quad \sum y^{2}=19,118 \quad \sum x y=3867 \end{aligned} $$ a. Do the data suggest the existence of a positive linear relationship (one in which an increase in \(y\) tends to be associated with an increase in \(x\) )? Test using a \(.05\) significance level. b. Predict flood damage resulting from a claim made when depth of flooding is \(3.5 \mathrm{ft}\), and do so in a way that conveys information about the precision of the prediction.

A sample of \(n=10,000(x, y)\) pairs resulted in \(r=\) .022. Test \(H_{0}: \rho=0\) versus \(H_{a}: \rho \neq 0\) at significance level .05. Is the result statistically significant? Comment on the practical significance of your analysis.

The accompanying figure is from the article "Root and Shoot Competition Intensity Along a Soil Depth Gradient" (Ecology [1995]: 673-682). It shows the relationship between above-ground biomass and soil depth within the experimental plots. The relationship is described by the linear equation: biomass \(=-9.85+\) \(25.29\) (soil depth) and \(r^{2}=.65 ; P<0.001 ; n=55\). Do you think the simple linear regression model is appropriate here? Explain. What would you expect to see in a plot of the standardized residuals versus \(x\) ?

The article "Technology, Productivity, and Industry Structure" (Technological Forecasting and Social Change [1983]: \(1-13\) ) included the accompanying data on \(x=\) research and development expenditure and \(y=\) growth rate for eight different industries. $$ \begin{array}{rrrrrrrrr} x & 2024 & 5038 & 905 & 3572 & 1157 & 327 & 378 & 191 \\ y & 1.90 & 3.96 & 2.44 & 0.88 & 0.37 & -0.90 & 0.49 & 1.01 \end{array} $$ a. Would a simple linear regression model provide useful information for predicting growth rate from research and development expenditure? Use a .05 level of significance. b. Use a \(90 \%\) confidence interval to estimate the average change in growth rate associated with a 1 -unit increase in expenditure. Interpret the resulting interval.

Exercise \(13.16\) described a regression analysis in which \(y=\) sales revenue and \(x=\) advertising expenditure. Summary quantities given there yield $$ n=15 \quad b=52.27 \quad s_{b}=8.05 $$ a. Test the hypothesis \(H_{0}: \beta=0\) versus \(H_{a}: \beta \neq 0\) using a significance level of \(.05 .\) What does your conclusion say about the nature of the relationship between \(x\) and \(y\) ? b. Consider the hypothesis \(H_{0}: \beta=40\) versus \(H_{a}: \beta>40\). The null hypothesis states that the average change in sales revenue associated with a 1 -unit increase in advertising expenditure is (at most) \(\$ 40,000\). Carry out a test using significance level .01.
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