Problem 3 (20 Points): In An Attempt To Determine Whether

Problem 3. (20 points) In an attempt to determine whether or not special

In this assignment, we examine two statistical problems involving hypothesis testing and regression analysis. The first problem investigates whether specialized training enhances the speed of assembly line workers at AMTEL Inc., by analyzing the improvement in their performance. The second problem involves modeling quarterly dishwasher sales using multiple regression to account for seasonal patterns and trend over time. The third problem tests whether a sample of high school classmates earns more than the national average salary for recent graduates. Each problem is approached systematically using the 7-step hypothesis testing or regression analysis procedure, including data coding, model estimation, significance testing, and interpretation.

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Problem 1: Effect of Special Training on Assembly Line Worker Performance

To evaluate whether specialized training leads to an improvement in the assembly time, we formulate hypotheses regarding the mean difference in improvement times before and after training. The null hypothesis (H₀) asserts that the training has no effect, implying that the mean improvement μ equals zero. Conversely, the alternative hypothesis (H₁) suggests that the training does result in an improvement, thus μ > 0. Given the sample data—25 workers with a mean improvement of 2.6 minutes and a standard deviation of 4 minutes—we perform a one-sample t-test at an unspecified significance level, typically α=0.05, to test these hypotheses.

Calculating the test statistic involves the formula t = (̄x - μ₀) / (s / √n), where ̄x = 2.6, μ₀ = 0, s=4, and n=25. This yields t = (2.6 - 0) / (4 / √25) = 2.6 / (4/5) = 2.6 / 0.8 = 3.25. The degrees of freedom are df = n - 1 = 24. Using a t-distribution table or software, we find the p-value corresponding to t=3.25 with 24 df. Since t=3.25 is likely to produce a p-value less than 0.05, we reject the null hypothesis, concluding that the training has statistically significantly improved the workers’ assembly times.

Therefore, based on the analysis, we infer that special training positively affects the speed at which assembly line workers complete their tasks, supporting the hypothesis that training enhances efficiency.

Problem 2: Modeling Quarterly Dishwasher Sales with Multiple Regression

The second problem involves analyzing quarterly dishwasher sales over two years, recognizing the presence of trend and seasonal fluctuations. The variables defined include the sales volume (Y), time variable (t), and seasonal dummy variables Q2, Q3, and Q4, with Q1 as the baseline category. Coding the data involves assigning t values sequentially from 1 to 8 for the eight quarters from 2010 to 2011. The dummy variables are coded as binary indicators: 1 if the quarter matches the specific dummy, 0 otherwise. This coding process facilitates estimating the influence of each seasonal period relative to the baseline.

Using Excel, the multiple regression model is fitted to the data: Y = β₀ + β₁ t + β₂ Q2 + β₃ Q3 + β₄ Q4 + ε. Suppose the regression results yield estimated coefficients as follows: β₀ (intercept), β₁ (trend), and seasonal coefficients β₂, β₃, β₄. The estimated regression equation may look like: Y = 150 + 3.5 t + 25 Q2 + 30 Q3 + 20 Q4, indicating how sales change over time and across quarters.

Interpreting the slopes of dummy variables: β₂ (= 25) suggests that sales during Q2 are, on average, $25 higher than Q1, all else equal; β₃ (= 30) indicates a $30 increase during Q3; β₄ (= 20) reflects a $20 rise in Q4 sales relative to Q1. These parameters measure seasonal effects relative to the baseline quarter.

To evaluate the overall fit of the model, hypotheses are formulated: H₀: All regression coefficients (except the intercept) are zero (no relationship), versus H₁: At least one coefficient is non-zero. An F-test for significance, based on the regression output, assesses whether the model explains a significant proportion of variance. If the F-statistic exceeds the critical value or the p-value is less than 0.05, H₀ is rejected, indicating the model provides a good fit to the data.

The bonus question involves examining individual significance: t-tests on each coefficient determine if specific variables significantly influence sales. Variables with p-values less than 0.05 are deemed significant; others are not. Such analysis helps refine the model by confirming important predictors and discarding redundant ones.

Problem 3: Testing if High School Classmates’ Salaries Exceed the National Average

The third problem analyzes whether a sample of recently graduated high school classmates earns more than the national average salary of $53,500 in 2015. The sample mean salary is X̄ = $56,975, with a standard deviation s= $5,210, and sample size n (not specified but necessary for calculation). The hypothesis test is H₀: μ = $53,500 versus H₁: μ > $53,500, using a one-sample t-test given the assumption of normality.

The test statistic is calculated as t = (X̄ - μ₀) / (s / √n). Assuming n is provided or estimated, we insert the values and compute t. The degrees of freedom are df = n - 1. If the calculated t exceeds the critical t (corresponding to α=0.05 or other level), we reject H₀ and conclude that their average salary is statistically significantly higher than the national average. Otherwise, we fail to reject the null hypothesis, suggesting no significant difference.

This analysis provides insight into whether the high school classmates are performing above the national average salary for recent graduates, taking into account sampling variability and the assumption of normality.

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