Problem 10 - Basic Statistics for Business and Economics with Student

Question

The manager of a computer software company wishes to study the number of hours senior executives by type of industry spend at their desktop computers. The manager selected a sample of five executives from each of three industries. At the .05 significance level, can she conclude there is a difference in the mean number of hours spent per week by industry?

Step-by-Step Solution

Answer

Conduct an ANOVA test and if the calculated F-statistic exceeds the critical F-value, conclude that there is a significant difference.

Step 1: State the Hypotheses

First, we set up the null and alternative hypotheses. The null hypothesis \(H_0\) states that there is no difference in the mean number of hours spent per week by industry, i.e., \( \mu_1 = \mu_2 = \mu_3 \). The alternative hypothesis \(H_a\) states that at least one industry's mean is different: \( \mu_i eq \mu_j \) for at least one pair \((i, j)\).

Step 2: Choose the Significance Level

We are given a significance level \( \alpha = 0.05 \). This means we will be making our decision of whether to reject the null hypothesis based on this threshold.

Step 3: Organize the Data and Calculate Means

Assume we have the sample data for each industry. Calculate the sample mean for each group of executives in their respective industries, \( \bar{x}_1, \bar{x}_2, \bar{x}_3 \).

Step 4: Conduct ANOVA Test

To determine if there's a statistically significant difference among the group means, conduct a one-way ANOVA. Calculate the F-statistic using:1. Between-group variation (SSB)2. Within-group variation (SSW)3. Total variation (SST = SSB + SSW)Then calculate the mean square for between-groups (MSB = SSB/dfB) and within-groups (MSW = SSW/dfW). The F-statistic is computed as \( F = \frac{MSB}{MSW} \).

Step 5: Decision Rule

Obtain the critical F-value from the F-distribution table using \(dfB\) and \(dfW\) as degrees of freedom, and the given significance \( \alpha = 0.05 \). Compare the calculated F-statistic to this critical value.

Step 6: Draw Conclusion

If the calculated F-statistic is greater than the critical F-value from the table, reject the null hypothesis, suggesting there is a significant difference in the mean hours spent by industry. Otherwise, do not reject the null hypothesis.

Key Concepts

Understanding Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics that helps us make decisions based on data. It starts by defining two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. In our example, it suggests there is no difference in the mean hours spent by executives across industries. The alternative hypothesis, \(H_a\), suggests there is a difference in the means for at least one pair of industries.

When conducting hypothesis testing, we are trying to decide whether to reject the null hypothesis in favor of the alternative hypothesis. The process involves gathering data, calculating a test statistic, and comparing it to a threshold. This threshold is determined by the chosen significance level. Hypothesis testing not only helps us determine if there is a difference in data groups, but also quantifies the reliability of this decision.

Defining the Significance Level

The significance level, often denoted by \( \alpha \), is an essential part of hypothesis testing. It sets the boundary for deciding when a result is statistically significant. In our case, a significance level of 0.05 means we accept a 5% chance of incorrectly rejecting the null hypothesis. In other words, there's a 5% risk that we might say there is a difference in means when there actually isn't one.

The choice of significance level impacts the balance between two types of errors: Type I and Type II. A lower \( \alpha \) reduces the risk of Type I error (false positive) but increases the risk of Type II error (false negative), and vice versa. Therefore, choosing an appropriate significance level is crucial as it affects the sensitivity and specificity of the test.

Exploring the F-statistic

In ANOVA (Analysis of Variance), the F-statistic plays a crucial role. It helps quantify the ratio of systematic variance to unsystematic variance. Simply put, the F-statistic measures how much of the variance in the data is explained by differences between group means compared to variance within groups.

The F-statistic is calculated by dividing the Mean Square for Between-groups (MSB) by the Mean Square for Within-groups (MSW). If the between-group variation is large compared to the within-group variation, the F-statistic will be high, pointing towards a significant difference.

In hypothesis testing, this statistic is then compared to a critical value from the F-distribution at the specified significance level. If the F-statistic exceeds the critical value, we reject the null hypothesis.

Understanding Between-group Variation

Between-group variation refers to the variation in data explained by the differences between group means. In ANOVA, it captures the extent to which the means of different groups differ. This is in contrast to within-group variation which describes how much the individual data points in each group deviate from their respective group means.

Identifying between-group variation is key in hypothesis testing as it shows the presence of systematic differences. If there is minimal between-group variation, it suggests that the groups are similar with respect to the measured variable. High between-group variation, on the other hand, indicates significant differences.

• Calculate the Sum of Squares Between (SSB) to quantify between-group variation.
• The Mean Square Between (MSB) is computed by dividing SSB by its degrees of freedom.

In summary, analyzing between-group variation allows us to determine if the differences across groups are substantial enough to be considered statistically significant.