Problem 4 - Basic Statistics for Business and Economics with Student

Question

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?

Step-by-Step Solution

Answer

No significant difference between the means at the 0.05 level.

Step 1: Define Hypotheses

Set up the null and alternative hypotheses.- Null Hypothesis \( H_0 \): There is no difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees, i.e., \( \mu_1 = \mu_2 \).- Alternative Hypothesis \( H_1 \): There is a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees, i.e., \( \mu_1 eq \mu_2 \).

Step 2: Identify Known Values

From the problem statement, extract the known values:- Sample mean for Cincinnati \( \bar{x}_1 = 370 \)- Sample mean for Pittsburgh \( \bar{x}_2 = 380 \)- Population standard deviation for Cincinnati \( \sigma_1 = 30 \)- Population standard deviation for Pittsburgh \( \sigma_2 = 26 \)- Sample sizes: \( n_1 = 35 \), \( n_2 = 40 \)- Significance level \( \alpha = 0.05 \)

Step 3: Calculate the Standard Error

The standard error of the difference between two means is calculated using the formula:\[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]Substitute the known values:\[ SE = \sqrt{\frac{30^2}{35} + \frac{26^2}{40}} = \sqrt{\frac{900}{35} + \frac{676}{40}} \approx \sqrt{25.71 + 16.9} \approx \sqrt{42.61} \approx 6.53 \]

Step 4: Calculate Test Statistic

The test statistic is calculated using the formula:\[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{SE} \]Since \( (\mu_1 - \mu_2) = 0 \), the equation simplifies to:\[ z = \frac{370 - 380}{6.53} = \frac{-10}{6.53} \approx -1.53 \]

Step 5: Determine Critical Value and Make Decision

For a two-tailed test at the 0.05 significance level, the critical values for \( z \) are \( -1.96 \) and \( 1.96 \). Compare the calculated test statistic with the critical values.- Since \( z = -1.53 \) does not fall in the critical region \((-\infty, -1.96] \cup [1.96, \infty)\), we fail to reject the null hypothesis.

Step 6: Conclusion

Based on the calculations, there is insufficient evidence at the 0.05 significance level to conclude that there is a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees.

Key Concepts

Corporate Employees

Understanding the behaviors and habits of corporate employees can be insightful, especially when examining differences across various locations. In the scenario given, we are comparing two distinct groups of employees located in Cincinnati and Pittsburgh. Both groups belong to the same corporation, PNC, Inc., but they operate in different environments. These individuals may have varying commuting experiences due to factors like public transportation options, the geographical layout of each city, and workplace policies.

When conducting such a study, it helps human resources and managers gauge the factors that might affect their employees’ jobs and day-to-day lives. This knowledge can aid in decision-making processes related to workplace flexibility, compensation benefits, and employee retention strategies. Understanding the mileage traveled can also contribute to planning logistical support or evaluating environmental impacts.

Population Standard Deviation

In statistics, the population standard deviation is a measure that indicates the amount of variation or dispersion in a population dataset. When conducting a hypothesis test, knowing the population standard deviation helps determine the variability among the data points across the entire population group.

In our corporate employee study, the population standard deviation values are provided for both Cincinnati (30 miles) and Pittsburgh employees (26 miles). These values suggest how spread out their monthly travel distances tend to be. The smaller the standard deviation, the more concentrated the data points are around the mean. With Pittsburgh having a slightly smaller population standard deviation, it implies their employees have more consistent travel distances compared to those in Cincinnati.

• This measurement is crucial when comparing the two groups since it feeds into the calculation of the standard error, helping us understand the stability of our sample means.

Sample Mean

The concept of a sample mean is vital in hypothesis testing as it provides an estimate of the true mean of a population. It is calculated by summing up all the sample values and dividing by the number of values. In simpler terms, it tells us what the 'average' observation is for the sample.

In our case, the sample means are given as 370 miles for Cincinnati and 380 miles for Pittsburgh. These sample means represent the central tendency of the monthly distances traveled by employees in the respective cities within the chosen samples.

By comparing the sample means, we aim to determine if there is a significant difference in the commuting habits of each city's employees. It is essential to note that these sample means are estimates of the population means; thus, they come with variability, which is accounted for with statistical measures like the standard error.

Standard Error

The standard error (SE) is a statistical term that measures the accuracy with which a sample represents a population. It is essentially the standard deviation of the sampling distribution of a statistic, most commonly the mean. In hypothesis testing, the standard error helps determine how far the sample mean is from the population mean.

The formula for calculating the standard error of the difference between two means incorporates the population standard deviations and the sample sizes: \[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \] In our example, substituting known values gives us an SE of approximately 6.53. This tells us how much "typical" variation we might expect if we were to draw different samples.

• A smaller standard error indicates that the sample mean is more precise, while a larger one suggests greater variability and less precision.

Understanding and calculating the standard error is a key step in hypothesis testing, as it plays a pivotal role in constructing test statistics and making inferences about the population means.