Adm 2304x Assignment 2 By Professor Afshin Kamyabnia ✓ Solved

Adm 2304x Assignment 2professor Afshin Kamyabniyata Total Marks

Analyze and perform statistical tests based on the provided datasets related to small business growth, MBA graduate salaries, beer bitterness, and student reading habits. The assignment involves hypothesis testing (including choosing appropriate tests, stating hypotheses, conditions, and interpreting results), calculating confidence intervals, creating graphical representations such as boxplots, and assessing the appropriateness of statistical methods. All work should include manual calculations and software outputs, with explanations linking hypotheses tests and confidence intervals. Specific datasets to be used are in the files Assign2Data.xlsx, MBA_Salary, Beer Bitterness, and related datasets, focusing on real-world examples for each statistical analysis task.

Paper For Above Instructions

Introduction

This comprehensive analysis addresses multiple statistical evaluation tasks, emphasizing hypothesis testing, confidence interval estimation, graphical representation, and methodological appropriateness across diverse real-world contexts, including small business growth, graduate salaries, beer bitterness, and reading habits among students. The assessment integrates manual calculations with software tools such as MS Excel, fostering a thorough understanding of statistical principles and their applications in managerial and social sciences.

Question 1: Impact of Canada Small Business Financing Program (CSBFP) on Firm Growth

Part a: Hypothesis Testing for Increase in Firm Size

The first task involves testing whether small IT firms in Ottawa that received CSBFP support experienced growth in number of employees from 2014 to 2018. A paired samples t-test is appropriate because data are paired, comparing the same firms across two time points.

Approach: Formulate null hypothesis (H0): no increase in mean number of employees; alternative hypothesis (Ha): mean number of employees increased in 2018 compared to 2014. Conditions for a t-test include (1) paired data, (2) continuous interval/ratio data, (3) approximately normal differences or sufficiently large sample size for the Central Limit Theorem.

Graphical check: Construct a histogram or boxplot of the differences in employees (2018 - 2014) to assess normality. The boxplot indicates [description based on graph], supporting the use of a t-test.

Calculate the mean difference, standard deviation, and use the t-distribution to compute the test statistic. Compare the p-value with α=0.05; reject H0 if p

Part b: Confidence Interval for Mean Increase

Construct a 95% confidence interval manually: CI = mean difference ± t(standard error), where t is the critical value from the t-distribution with n-1 degrees of freedom. If the interval does not include zero, conclude that there is a significant increase in firm size.

Part c: Relationship Between Confidence Interval and Hypothesis Testing

The confidence interval provides a range of plausible values for the mean increase. If zero is not within this interval, it aligns with rejecting H0 in the hypothesis test, supporting the conclusion of firm growth.

Question 2: Salary Differences Between MBA Graduates of 2015 and 2016

Part a: Boxplot Comparison

Create side-by-side boxplots for the two cohorts using salary data. Observe median differences, variability, and potential outliers to understand distributional differences.

Part b: Non-parametric Test (Mann-Whitney U test)

Hypotheses:

H0: Salaries for 2015 graduates are not higher than for 2016 graduates.

Ha: Salaries for 2015 graduates are higher than for 2016 graduates.

Since normality may not be guaranteed, Mann-Whitney U test is appropriate.

Manual calculations involve ranking combined data, summing ranks for the group, computing U statistic, and comparing with critical U at α=0.05.

Part c: Software Comparison

Use Excel's Data Analysis add-in or another tool to perform the Mann-Whitney U test. Compare software output with manual results to ensure consistency and discuss any discrepancies.

Question 3: Comparing Beer Bitterness (IBU) in Two Beer Types

Part a: Boxplot Examination

Create boxplots for Midnight Ottawa and Midnight Montreal bitterness levels. Visual analysis suggests [description], indicating distribution shape and variability.

Part b: Difference in Mean Bitterness (Unequal Variances)

Perform Welch's t-test with the hypotheses:

H0: No difference in mean bitterness.

Ha: There is a difference.

Calculate means, variances, sample sizes, and the test statistic, then determine the p-value.

Part c: 99% Confidence Interval for Difference

Compute the confidence interval using the formula: difference of sample means ± t* × standard error, accounting for unequal variances.

Part d: Reflection on Hypothesis Test and Confidence Interval

The hypothesis test assesses whether a significant difference exists, while the confidence interval estimates the magnitude and direction of the difference. Both methods are interconnected—if zero is outside the confidence interval, the test likely rejects H0.

Part e: Assuming Equal Variances

Repeat the test assuming equal variances, using pooled variance for calculations. Show manual computation steps and interpret whether conclusions align with unequal variance assumptions.

Question 4: University Reading Trends and Chi-Squared Test

Part a: Data Table Construction

Organize observed counts in a contingency table: rows for book categories, columns for universities. Compute expected counts assuming independence, using row and column totals divided by grand total, formatted to three decimal places.

Part b: Chi-Squared Test of Independence

State hypotheses:

H0: Distribution of books read is independent of university.

Ha: Distribution depends on university.

Calculate chi-squared statistic: sum of (observed - expected)^2 / expected over all cells, compare to critical value at α=0.01, interpret significance.

Part c: Appropriateness of Chi-Squared Test

Assess whether expected counts satisfy the assumptions for the chi-squared test (>5 in most cells). If not, suggest alternative methods or caution in interpretation.

Conclusion

This assignment integrates hypothesis testing, confidence interval estimation, graphical analysis, and evaluation of statistical methods within real-world datasets. Accurate manual calculations paired with software outputs bolster understanding of statistical inference, relevant in managerial decision-making and social science research.

References

• Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson.
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
• Glass, G. V. (1976). Primary, secondary, and meta-analysis of research. American Educational Research Journal, 13(4), 351-379.
• Kirk, R. E. (2013). Experimental Design: Procedures for the Behavioral Sciences. Sage.
• McDonald, J. H. (2014). Handbook of Biological Statistics. Sparky House Publishing.
• Motulsky, H. (2014). Intuitive Biostatistics. Oxford University Press.
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
• Zar, J. H. (2010). Biostatistical Analysis. Pearson.
• Urdan, T. (2017). Statistics in Plain English. Routledge.