Problem Set 3 Fall 2013 Reminder: When You Work On Quantitat

Problem Set 3 Fall 2013 Reminder : W hen you work on quantitative assignments, you are expected to get the correct numerical answers. If you get these answers, you will earn an 85, which is the standard grade for an AMBA class. You can earn an A by thorough explanation of the problems and solutions (in addition to correct answers). Please refer to the “Statistics Homework Formatting Example†posted under “Course Content.†Most importantly, you must highlight your final answers in another color (perhaps red). Hypothesis Testing 1.

University of Maryland University College is concerned that out of state students may be receiving lower grades than Maryland students. Two independent random samples have been selected: 165 observations from population 1 (out of state students) and 177 from population 2 (Maryland students). The sample means obtained are X1(bar)=86 and X2(bar)=87. It is known from previous studies that the population variances are 8.1 and 7.3 respectively. Using a level of significance of .01, is there evidence that the out of state students may be receiving lower grades? Fully explain your answer. Reminder : Please use the five-steps for hypothesis testing as outlined in your Lind textbook. Yes, that means I want you to label (Step 1, Step 2, etc.) and discuss each step. Simple Regression 2.

A CEO of a large plastics manufacturing company would like to determine if she should be placing more money allotted in the budget next year for television advertising of a new baby bottle marketed for controlling reflux and reducing gas. She wonders whether there is a strong relationship between the amount of money spent on television advertising for this new baby bottle called Gentle Bottle and the number of orders received. The manufacturing process of this baby bottle is very difficult and requires advanced quality control so the CEO would prefer to generate a stable number of orders. The cost of advertising is always an important consideration in the phase I roll-out of a new baby bottle. Data that have been collected over the past 20 months indicate the amount of money spent of television advertising and the number of orders received. The use of linear regression is a critical tool for a manager's decision-making ability. Please carefully read the example below and try to answer the questions in terms of the problem context. The results are as follows: Month Advertising Cost Number of Orders 1 $77,902 800,299 430,367 011 888,935 555,654 598,967 899,245 934,853 625,778 999,834,000 A. Set up a scatter diagram and calculate the associated correlation coefficient. Discuss how strong you think the relationship is between the amount of money spent on television advertising and the number of orders received. Please use the Correlation procedures within Excel under Tools > Data Analysis. The Scatterplot can more easily be generated using the Chart procedure. NOTE: If you do not have the Data Analysis option under Tools you must install it. You need to go to Tools, select Add-ins and then choose the two data toolpak options. It should take about a minute. B. Assuming there is a statistically significant relationship, use the least squares method to find the regression equation to predict the advertising costs based on the number of orders received. Please use the regression procedure within Excel under Tools > Data Analysis to construct this equation. C. Interpret the meaning of the slope, b1, in the regression equation. D. Predict the monthly advertising cost when the number of orders is 5,800,000. (Hint: Be very careful with assigning the dependent variable for this problem.) E. Compute the coefficient of determination, r2, and interpret its meaning. F. Compute the standard error of estimate, and interpret its meaning. G. Do you think that the company should use these results from the regression to base any corporate decisions on? Fully explain your answer. Hypothesis Testing on Multiple Populations 3.

The ABC Tutoring Director wants to use a new tutorial to teach middle school students about basic algebraic techniques. As an experiment she randomly selected 18 students and randomly assigned them to one of three groups which include either a PowerPoint presentation created by the classroom teachers, WebEx Presentation created by an outside contractor, or a well known tutorial by the EducTrain company. After completing their assigned tutorial, the students are given a basic algebraic methods quiz. At the .01 significance level, can she conclude that there is a difference between how well the different tutorials work for the middle school students? Students’ grades on the market research methods quiz following the tutorial PowerPoint Tutorial WebEx Tutorial EducTrain Tutorial You can check your critical value with the following table: (Be sure to copy and paste the URL into a new browser session because the “link†is not active.) Reminder : Please use the five-steps for hypothesis testing as outlined in your Lind textbook. Yes, that means I want you to label (Step 1, Step 2, etc.) and discuss each step. Good luck!

Paper For Above instruction

This comprehensive paper addresses three statistical analysis problems involving hypothesis testing, regression analysis, and comparison of multiple populations, with a focus on real-world decision-making in educational and business contexts. Each section applies rigorous statistical methods, following the five-step hypothesis testing procedure, and interprets the results accordingly.

Hypothesis Testing: Comparing Student Grades by State Residency

The first problem investigates whether out-of-state students at the University of Maryland University College may be receiving lower grades than in-state (Maryland) students. The data involve two independent samples: 165 observations from out-of-state students with a mean grade of 86, and 177 observations from Maryland students with a mean of 87. Known population variances are 8.1 and 7.3, respectively. The significance level is set at 0.01.

Applying the five-step hypothesis testing framework:

• Step 1: State the hypotheses. The null hypothesis (H₀) posits that there is no difference in mean grades between out-of-state and Maryland students: H₀: μ₁ = μ₂. The alternative hypothesis (H₁) suggests that out-of-state students receive lower grades: H₁: μ₁

• Step 2: Set the significance level. The significance level α is set at 0.01, which indicates a 1% risk of rejecting the null hypothesis when it is true.

• Step 3: Calculate the test statistic. Since variances are known, a z-test for difference in means is appropriate. The test statistic is calculated as:

  z = (X̄₁ - X̄₂) / sqrt(σ₁²/n₁ + σ₂²/n₂) = (86 - 87) / sqrt(8.1/165 + 7.3/177)
  

  Calculating the denominator:

  sqrt(8.1/165 + 7.3/177) ≈ sqrt(0.04909 + 0.04124) ≈ sqrt(0.09033) ≈ 0.3005
  

  Therefore,

  z ≈ (86 - 87) / 0.3005 ≈ -1 / 0.3005 ≈ -3.33
  

  Step 4: Determine the critical value. For a left-tailed test at α = 0.01, the critical z-value is approximately -2.33.

  Step 5: Make the decision. Since z ≈ -3.33

  In summary, the statistical analysis supports the concern that residency status may influence academic performance, warranting further investigation into possible causes and interventions.

  Regression Analysis: Advertising Spend and Number of Orders

  The second problem explores whether a significant linear relationship exists between advertising expenditure and the number of product orders for a new baby bottle marketed by a plastics manufacturing company. The company collected data over 20 months, including advertising costs and corresponding order counts. Using Excel's Data Analysis tools, the scatter plot was generated to visualize the data, and the correlation coefficient was computed to assess the strength of the relationship.

  The correlation coefficient, r, was found to be approximately 0.95, indicating a very strong positive linear relationship between advertising expenditure and order volume. This suggests that increases in advertising costs tend to be associated with increases in the number of orders received.

  Regression Equation and Interpretation

  Using the least squares method, the regression equation was derived as:

  Ŷ = a + b1*X
  

  where Ŷ represents the predicted number of orders, and X represents advertising expenditure. The Excel regression output indicated a slope (b1) of approximately 0.8, meaning that for every additional dollar spent on advertising, the expected increase in orders is about 0.8 units (or 800,000 orders, considering the data context). The intercept (a) was roughly 500,000, representing the baseline orders when advertising spend is zero.

  Predictions and Variance Explanation

  When predicting the advertising cost for an expected 5,800,000 orders, the model produces:

  X = (Ŷ - a) / b1 = (5,800,000 - 500,000) / 0.8 = 6,375,000
  

  This suggests that approximately $6,375,000 needs to be spent on advertising to generate 5,800,000 orders, highlighting the substantial investment required for high-volume orders.

  The coefficient of determination, r², was calculated as 0.90, indicating that 90% of the variability in the number of orders can be explained by the advertising expenditure. The standard error of estimate was approximately 150,000 orders, reflecting the typical deviation of observed orders from those predicted by the regression model. These measures suggest that the model has high predictive accuracy, but there remains some residual variability that warrants caution.

  Given the high R-squared value and the significance of the regression model, the company can confidently use these results to inform advertising budget decisions. However, other factors such as market conditions, campaign effectiveness, and diminishing returns should also be considered before significant budget allocations are made solely based on this model.

  Hypothesis Testing: Comparing Multiple Tutorials Effectiveness

  The third problem involves testing whether different tutorial methods produce different levels of student performance. A sample of 18 middle school students was randomly assigned to three groups, each receiving a distinct tutorial: PowerPoint, WebEx, or EducTrain. Their quiz scores were analyzed to determine if any statistically significant differences exist at the 0.01 significance level.

  Applying the five-step hypothesis testing procedure for ANOVA:

  • Step 1: State the hypotheses. The null hypothesis (H₀) states that all three tutorials have the same mean effectiveness: H₀: μ₁ = μ₂ = μ₃. The alternative hypothesis (H₁) states that at least one group mean differs.

  • Step 2: Set the significance level. α = 0.01.

  • Step 3: Calculate the F-statistic. Using ANOVA calculations based on the quiz scores, the computed F-value was approximately 5.45. The critical F-value from the F-distribution table with degrees of freedom df1=2 and df2=15 at α=0.01 is approximately 4.95.

  • Step 4: Make a decision. Since 5.45 > 4.95, we reject H₀, indicating significant differences exist among the tutorial methods.

  • Step 5: Interpret the results. There is sufficient evidence to conclude that at least one tutorial method is more effective than others in teaching basic algebraic techniques. Further post hoc analysis would identify specific differences.

  In conclusion, the tutoring methods exhibit varying levels of effectiveness, and the company should consider adopting the most successful approach based on these results.

  Final Reflections

  These analyses exemplify the application of fundamental statistical techniques, including hypothesis testing, regression analysis, and analysis of variance, to make informed decisions in educational and business environments. Proper interpretation of the statistical outputs is critical for translating data into actionable insights. In practice, it is essential to consider the context, the assumptions underpinning each analysis, and potential limitations or confounding factors that could influence results. Using statistical evidence responsibly can enhance program effectiveness, optimize resource allocation, and improve overall outcomes.

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