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Using the data provided, this assignment requires formulating hypotheses, performing statistical calculations such as test statistics and p-values, and constructing confidence intervals for various scenarios. The objective is to apply hypothesis testing and confidence interval techniques using Excel, supported by appropriate data organization and formulas.

Paper For Above instruction

Introduction

Statistical analysis forms a crucial part of decision-making processes across various fields, including finance, political science, marketing, and economics. This paper addresses three statistical problems, emphasizing the formulation of hypotheses, calculation of test statistics, and interpretation via p-values and confidence intervals. The utilization of Excel for data organization and computational purposes is central to this approach, as it facilitates precise and efficient analysis.

Problem 1: Hypothesis Testing for a Population Mean

Given a sample of 81 account balances with a mean of $1,200 and a standard deviation of $126, the task is to determine whether the population mean differs significantly from $1,150. The hypotheses are:

  • Null hypothesis (H₀): μ = $1,150
  • Alternative hypothesis (H₁): μ ≠ $1,150

This represents a two-tailed test. Using Excel, the test statistic (z) can be calculated by:

z = (x̄ - μ₀) / (σ / √n)

Where:

  • x̄ = 1,200 (sample mean)
  • μ₀ = 1,150 (hypothesized mean)
  • σ = 126 (standard deviation)
  • n = 81 (sample size)

Calculated in Excel, this yields:

z = (1200 - 1150) / (126 / √81) = 50 / (126 / 9) = 50 / 14 = approximately 3.57

Using Excel's T.DIST or NORM.S.DIST, the two-tailed p-value can be found, which for z = 3.57 is approximately 0.00036. Since p

Problem 2: Comparing Proportions in Two States

In Alabama and Mississippi, pre-election polls indicate support for the Democratic candidate. Suppose in Alabama, 510 out of 1000 voters favor the candidate, and in Mississippi, 470 out of 950 do so. The hypotheses are:

  • H₀: p₁ = p₂ (proportions are equal)
  • H₁: p₁ ≠ p₂ (proportions are different)

Calculating the pooled proportion:

p̂ = (x₁ + x₂) / (n₁ + n₂) = (510 + 470) / (1000 + 950) = 980 / 1950 ≈ 0.5026

Excel formulas can compute the test statistic for difference in proportions:

z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]

where p̂₁ = 0.51, p̂₂ = 0.4947. Calculations in Excel will give a z-value close to 1.54, with a p-value approximately 0.123. Since p > 0.05, we fail to reject H₀, implying no significant difference in voter support between the two states at the 95% confidence level.

Problem 3: Confidence Interval and Testing for Income Differences

The data provided allows analyzing whether the average yearly incomes of marketing managers differ between the East and West U.S. regions. The sample means and standard deviations are used to develop confidence intervals and perform hypothesis tests.

The confidence interval for the difference in means with a 95% confidence level is derived using:

(x̄₁ - x̄₂) ± t* √(s₁²/n₁ + s₂²/n₂)

where:

  • x̄₁, s₁, n₁ = East region sample mean, standard deviation, and size
  • x̄₂, s₂, n₂ = West region sample mean, standard deviation, and size

Similar calculations in Excel will involve calculating the degrees of freedom using the Welch-Satterthwaite approximation, determining the critical t-value, and constructing the interval.

The hypothesis test assesses whether the mean incomes significantly differ from each other, with the null hypothesis being:

  • H₀: μ₁ - μ₂ = 0
  • H₁: μ₁ ≠ μ₂

Using the p-value method in Excel, the Z-test statistic or t-test statistic will be compared against the significance level (α=0.05). If p

Additional Data Analysis: ROI Comparison Between Business and Engineering Majors

In analyzing ROI data between majors, a two-sample t-test is performed. For Business majors, the average ROI is hypothesized to be less than that for Engineering majors at the 10% significance level. The hypotheses are:

  • H₀: μ₁ ≥ μ₂ (Business ROI is not less than Engineering ROI)
  • H₁: μ₁

Using Excel, the means, variances, and sample sizes are calculated for each major. The t-statistic is computed assuming unequal variances (Welch's t-test). The p-value for the one-tailed test determines whether to reject H₀. A p-value less than 0.10 supports the conclusion that Business majors' ROI is significantly less than that of Engineering majors.

Conclusion

Applying hypothesis testing and confidence intervals in Excel provides a rigorous approach to decision-making in various scenarios. Proper data organization, formula usage, and interpretation of p-values facilitate informed conclusions. The analysis illustrates the importance of statistical methods in business, politics, and economics, emphasizing the value of Excel tools in executing these analyses effectively.

References

  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • Hogg, R. V., Tanis, E. A., & Zimmerman, D. L. (2013). Probability and Statistical Inference (9th ed.). Pearson.
  • Levine, D. M., Stephan, D. F., Krehbiel, T. C., & Berenson, M. L. (2013). Statistics for Managers Using Microsoft Excel (6th ed.). Pearson.
  • Payscale.com. (2013). Best College ROI by Major. Payscale Inc.
  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
  • Vogt, W. P. (2011). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. Sage Publications.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Everitt, B. S. (2002). The Cam-bridge Dictionary of Statistics. Cambridge University Press.

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