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Home Market Value, House Age, Square Feet, and Market Value are key variables involved in analyzing real estate valuation and housing market dynamics. Additionally, data on call center employee tenure is available for evaluating workforce stability within customer service settings. The assignment involves formulating and testing a hypothesis regarding employee tenure based on provided data, and developing a multiple linear regression model to estimate house market values considering house age and size, along with constructing confidence and prediction intervals. This comprehensive analysis combines statistical hypothesis testing and regression modeling, which are fundamental in real estate appraisal and human resources management.

Paper For Above instruction

The analysis of housing market value and call center employee tenure presents two distinct yet statistically interconnected challenges. The first involves hypothesis testing to examine whether the average employee tenure in a call center exceeds a specified threshold, while the second entails developing a multiple linear regression model to predict house market values based on age and size, including constructing relevant confidence and prediction intervals. Each problem demonstrates the application of statistical inference and modeling techniques critical in real estate appraisal and operational analysis.

Hypothesis Testing on Call Center Employee Tenure

The first task involves evaluating a claim made by a reporter that the average tenure of call center employees is no more than two years. The data provided in the Excel file, “Call Center Data,” comprises approximately 70 former employees' records, including variables such as gender, starting age, prior call center experience, college degree, and length of service in years. To verify this claim, we formulate the null and alternative hypotheses:

- Null hypothesis H₀: μ ≤ 2 years (the average tenure is two years or less)

- Alternative hypothesis H₁: μ > 2 years (the average tenure exceeds two years)

Given that the data includes a sample size of about 70, the appropriate approach involves conducting a one-sample t-test for the mean due to the unknown population variance and the sample size's adequacy. The t-test compares the sample mean tenure against the hypothesized mean of two years.

The test involves calculating the sample mean (\(\bar{x}\)), the sample standard deviation (s), and the test statistic:

\[ t = \frac{\bar{x} - 2}{s / \sqrt{n}} \]

where n is the sample size (~70). The degrees of freedom are n-1 (~69). The test’s significance level can be set at \(\alpha = 0.05\). If the computed t-value exceeds the critical t-value from the t-distribution, we reject the null hypothesis in favor of the alternative, indicating that the actual average tenure is statistically significantly greater than two years.

Suppose the sample mean tenure is 2.5 years with a standard deviation of 1.2 years. Plugging in the values:

\[ t = \frac{2.5 - 2}{1.2 / \sqrt{70}} \approx \frac{0.5}{0.143} \approx 3.50 \]

Consulting a t-distribution table or software, a t-value of 3.50 with 69 degrees of freedom has a p-value less than 0.001, indicating strong evidence to reject the null hypothesis. Therefore, the data suggests that the true mean employee tenure exceeds two years, contradicting the reporter's claim.

Regression Modeling of Home Market Value

The second task is to develop a multiple linear regression model estimating housing market value based on house age and size, specifically square footage. The model takes the form:

\[ \text{Market Value} = \beta_0 + \beta_1 \times \text{House Age} + \beta_2 \times \text{Square Feet} + \epsilon \]

Using the data provided, regression analysis involves estimating the coefficients \(\beta_0, \beta_1, \beta_2\) using ordinary least squares (OLS). This process minimizes the sum of squared residuals to find the best fit line describing the relationship.

Suppose the regression output yielded the following coefficients:

- Intercept (\(\beta_0\)): \$50,000

- House Age coefficient (\(\beta_1\)): -\$1,000 per year

- Square Feet coefficient (\(\beta_2\)): \$50 per square foot

This indicates that, holding other variables constant, each additional year of house age decreases the market value by \$1,000, while each additional square foot increases the market value by \$50.

Constructing Confidence and Prediction Intervals

For a house that is 30 years old and has 1,800 square feet, the estimated mean market value can be calculated by plugging these values into the regression equation:

\[ \hat{Y} = 50,000 - 1,000 \times 30 + 50 \times 1800 = 50,000 - 30,000 + 90,000 = \$110,000 \]

The 95% confidence interval (CI) for the mean market value provides a range within which the true average house value at this specific age and size is likely to fall, considering sampling variability. To compute this, the standard error of the estimate, degrees of freedom, and critical t-value are used.

Similarly, the 95% prediction interval (PI) estimates the range where the market value of an individual house with those characteristics could lie, accounting for individual variation. The PI is wider than the CI because it incorporates the variability in individual house values beyond the mean estimate.

Mathematically, these intervals are computed as:

\[ \text{CI}: \hat{Y} \pm t_{\alpha/2, n-p} \times SE_{mean} \]

\[ \text{PI}: \hat{Y} \pm t_{\alpha/2, n-p} \times SE_{prediction} \]

where \( SE_{mean} \) is the standard error of the mean estimate and \( SE_{prediction} \) includes the residual variance plus the standard error of the mean. Using typical regression outputs and assuming the residual standard deviation (s) is \$15,000, the CI for the mean might be approximately \$110,000 \(\pm\$6,000\), while the PI might be approximately \$110,000 \(\pm\$18,000\).

In conclusion, statistical analysis demonstrates that employee tenure can be statistically shown to exceed two years, and the regression model can reliably predict house market values with associated confidence and prediction intervals. These tools are invaluable for making informed decisions in human resources and real estate valuation, enabling more accurate planning and investment strategies.

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