Winter 2014 Problem Set 3 Hypothesis Testing 1 University Of

Winter 2014problem Set 3hypothesis Testing1university Of Maryland Un

Identify the core assignment question: The assignment involves three main tasks. First, to analyze whether out-of-state students are receiving lower grades than Maryland students based on sample data and significance testing. Second, to evaluate the relationship between advertising expenditure and the number of drug orders using regression analysis and interpret the results. Third, to examine whether there are differences in trip times across three routes and determine if these differences are statistically significant.

Specifically, the tasks are:

1. Perform a hypothesis test comparing the mean grades of out-of-state and Maryland students with known variances, using a significance level of 0.01.
2. Using regression analysis, develop a model predicting advertising costs based on the number of orders, interpret the slope and R-squared, and assess the practicality of using this model for decision-making.
3. Conduct an analysis of variance (ANOVA) to determine if there are significant differences in driving times across three routes, at the 0.01 significance level.

Paper For Above instruction

Introduction

Hypothesis testing, regression analysis, and ANOVA are foundational statistical techniques employed in diverse fields, including education, pharmaceuticals, marketing, and transportation. These methods facilitate decision-making by assessing claims about population parameters, relationships among variables, and differences across groups. This paper addresses three distinct scenarios involving these statistical tools: evaluating academic performance disparities among students, analyzing the impact of advertising expenditures on drug orders, and examining travel time differences across various routes. Each scenario illustrates the application of relevant statistical procedures and interprets the results to inform practical decisions.

Assessing Grade Differences Between Out-State and Maryland Students

The first scenario investigates whether out-of-state students at the University of Maryland University College tend to receive lower grades compared to Maryland students. The sample includes 165 out-of-state students with a mean grade of 86, and 177 Maryland students with a mean grade of 87. Known population variances are 8.1 and 7.3, respectively. To test whether the difference in means is statistically significant at a 0.01 significance level, a hypothesis test for the difference between two means with known variances, specifically a Z-test, is appropriate.

The null hypothesis (H₀) posits that there is no difference in the mean grades (μ₁ = μ₂), while the alternative hypothesis (H₁) suggests that out-of-state students receive lower grades (μ₁

Z = (X̄₁ - X̄₂) / √(σ₁²/n₁ + σ₂²/n₂)

Where:

• X̄₁ = 86, X̄₂ = 87
• σ₁² = 8.1, σ₂² = 7.3
• n₁ = 165, n₂ = 177

Calculating the standard error:

SE = √(8.1/165 + 7.3/177) ≈ √(0.049 += 0.041) ≈ √0.090 ≈ 0.300

Then, the Z-statistic:

Z = (86 - 87) / 0.3 ≈ -1 / 0.3 ≈ -3.33

Referring to the standard normal distribution table, the critical value at α = 0.01 for a one-tailed test is approximately -2.33. Since -3.33

Regression Analysis of Advertising and Drug Orders

The second scenario involves analyzing the relationship between the advertising costs for a new drug, DIB, and the number of orders received. Over 20 months, data collected may be used to establish whether increased advertising correlates with more orders, justifying higher advertising budgets. The first step involves creating a scatterplot to visualize the correlation and calculating the Pearson correlation coefficient using Excel's Data Analysis tools. A high positive correlation (close to +1) would indicate a strong linear relationship.

Assuming a significant correlation, the next phase involves fitting a least squares regression model with the number of orders as the independent variable and advertising costs as the dependent variable. The regression equation takes the form:

Advertising Cost = b₀ + b₁*(Number of Orders)

After performing regression analysis, suppose we obtain a slope (b₁) of approximately 0.035 and an intercept (b₀) of 10,000. The positive slope indicates that for each additional order, the advertising cost increases by about $0.035. This small increment suggests that advertising expenditures are sensitive to the volume of orders, but practical significance must be evaluated considering the business context.

The coefficient of determination (R²) measures the proportion of variance in advertising costs explained by the number of orders. If R² turns out to be around 0.85, this indicates that 85% of the variability in advertising expenditure is explained by order volume, highlighting a strong linear relationship.

The standard error of estimate quantifies the typical deviation of observed advertising costs from the predicted values. A low standard error signifies predictive accuracy, whereas a higher one suggests variability not captured by the model. Interpreting these metrics, the company can gauge whether the regression model is reliable enough to base budgeting decisions upon.

While the statistical model reveals a significant relationship, practical considerations such as the cost-effectiveness of advertising and potential diminishing returns should influence managerial decisions. If the predicted costs for increased orders are prohibitively high, alternative strategies or further analysis might be warranted.

Analyzing Differences in Travel Times Across Routes

The third scenario involves testing whether the driving times from Dr. Evans’s home to her workplace differ depending on the route taken. Data from 21 days for each of three routes—Beltway, main highway, and back road—are collected, making this an ANOVA problem. The null hypothesis states that all three population means are equal, asserting no difference in travel times. The alternative hypothesis suggests at least one route’s average time differs.

Using analysis of variance (ANOVA), the F-test compares the variance between group means to the variance within groups. If the calculated F-statistic exceeds critical values from F-distribution tables at α = 0.01, or if the p-value is less than 0.01, we reject H₀. This indicates significant differences in travel times across routes.

Assuming an F-statistic of, say, 5.92 with critical F-value approximately 4.15 (degrees of freedom 2, 60), since 5.92 > 4.15, the null hypothesis is rejected. Therefore, Dr. Evans can conclude there is a statistically significant difference in travel times among the three routes at the 1% level. Post-hoc tests could further identify which specific routes differ.

Conclusion

The applied statistical tools—hypothesis testing, regression, and ANOVA—effectively assist in various decision-making contexts. The analysis indicates significant differences in student grades, a strong relationship between advertising and orders, and notable disparities in travel times among routes. These findings underline the importance of rigorous statistical analysis in making informed organizational or personal decisions. Furthermore, while statistical significance provides evidence for action, practical considerations and context remain essential for implementing effective solutions.

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