Deciding On Null And Alternative Hypotheses, Test Statistic

Deciding on the null and alternative hypotheses, test statistics, and interpretation

The assignment involves analyzing multiple statistical scenarios, including hypothesis testing for means and proportions, analysis of variance (ANOVA), correlation analysis, and regression analysis. The goal is to formulate hypotheses, determine decision rules, compute test statistics, and interpret the results based on sample data against specified significance levels.

Specifically, the tasks include testing claims about population means and proportions using z- and t-tests, performing ANOVA for comparing multiple groups, examining interactions in two-factor ANOVA, building regression models for predicting values, and evaluating relationships through correlation coefficients and scatter plots. The analysis requires understanding of statistical theory, calculations of test statistics, critical value comparisons or p-value assessments, and drawing appropriate conclusions about the hypotheses.

Paper For Above instruction

Statistical hypothesis testing and analysis are fundamental in making informed decisions based on data. This paper explores multiple scenarios involving hypothesis tests for means and proportions, analysis of variance (ANOVA), regression models, and correlation analysis, illustrating the application of statistical theory to real-world problems.

Hypothesis Testing for Mean Waiting Time

The first scenario involves testing if the mean waiting time at a hospital emergency department is less than 55 minutes. The population standard deviation is known, and a sample mean of 54 minutes was observed from 50 patients. The null hypothesis (H₀) presumes no change, i.e., μ = 55, while the alternative hypothesis (H₁) posits that the mean is less than 55. The significance level (α) is 0.025, indicating a 2.5% threshold for type I error.

The decision rule involves calculating the z-test statistic: z = (x̄ - μ₀) / (σ/√n), which equals (54 - 55) / (15/√50). If the computed z is less than the critical z-value (-1.96 for a one-tailed test at 0.025), the null hypothesis is rejected, supporting the claim that the mean waiting time is less than 55 minutes.

Calculations yield z ≈ -0.4714, which is not less than -1.96. Therefore, the null hypothesis is not rejected, indicating insufficient evidence to support a reduction in average waiting time based on this sample.

Testing Mean Diameter of Ball Bearings

The second task examines whether the mean diameter of ball bearings differs from 50 mm. With a sample of 100 bearings, a sample mean of 51.2 mm, and a sample standard deviation of 1.223 mm, the hypothesis test uses the sample standard deviation as an estimate for σ, which is acceptable for large samples. The null hypothesis states μ = 50, and the alternative (H₁) claims μ ≠ 50. The significance level is 0.02.

The test statistic is t = (x̄ - μ₀) / (s/√n) = (51.2 - 50) / (1.223/√100) ≈ 1.2 / 0.1223 ≈ 9.80. With degrees of freedom 99, the critical t-value for a two-tailed test at α = 0.02 is approximately ±2.33. Since 9.80 > 2.33, we reject the null hypothesis, indicating significant evidence that the mean diameter differs from 50 mm.

Proportion Hypothesis Test for TV Viewership

The third problem assesses if the CBS affiliate's claimed viewership proportion of 41% is supported by sample data where only 36% (36 out of 100) watch the newscast. Null hypothesis H₀: p = 0.41, alternative H₁: p ≠ 0.41, with α = 0.01. The decision involves calculating the z-test: z = (p̂ - p₀) / √[p₀(1 - p₀)/n], which equals (0.36 - 0.41) / √[0.41×0.59/100] ≈ -0.05 / 0.0499 ≈ -1.00. The critical z-values at α = 0.01 for a two-tailed test are ±2.58. Since -1.00 is within -2.58 and 2.58, we fail to reject H₀, concluding the evidence does not contradict the station's claim.

Comparing Average Savings with Z-Test

The fourth scenario compares average savings balances between two groups of depositors. Group one (less than 3 years): mean = $1200, SD = $100, n unknown; group two (more than 3 years): mean = $1250, SD = $250, sample size n= ? (assumed large or known). Assuming large samples and known standard deviations, the z-statistic is calculated as z = (mean₁ - mean₂) / √(SD₁²/n₁ + SD₂²/n₂). Without explicit sample sizes, the calculation cannot proceed precisely. If n values are provided, the researcher would substitute these to test for significance at a specified alpha level.

Testing Bacteria Levels in Soil Samples

The fifth case involves comparing proportions of contaminated samples from Lake Erie and Lake Superior. Lake Erie: 70 contaminated out of 100 samples (p₁=0.70); Lake Superior: 150 samples, 90 contaminated (p₂=0.60). The null hypothesis H₀: p₁ = p₂. The pooled proportion p̂ = (70 + 90) / (100 + 150) = 160/250 = 0.64. The test statistic is z = (p₁ - p₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)] ≈ (0.70 - 0.60) / √[0.64×0.36(1/100 + 1/150)] ≈ 0.10 / 0.0708 ≈ 1.41. Critical z-value at α=0.05 (two-tailed) is ±1.96. Since 1.41

Difference in Customer Satisfaction Pre- and Post-Training

The sixth scenario involves a paired t-test with seven customers, pre- and post-training survey scores. The data are paired, and the test compares the mean differences. The t-statistic is calculated as t = (mean difference) / (standard deviation of differences / √n). Precise differences per customer are needed for calculation. Once differences (d_i) are computed, their mean and standard deviation give the numerator and denominator for t, which is then compared to critical t-value (df=6) at α=0.05.

ANOVA on Fuel Economy of Different Gasoline Grades

The seventh problem involves conducting a one-way ANOVA to compare miles per gallon across four gasoline grades. The null hypothesis states all group means are equal, and the alternative suggests at least one differs. The ANOVA table includes between-group variance, within-group variance, degrees of freedom, and F-statistic. If the computed F exceeds the critical F-value from F-distribution tables at α=0.05, the null is rejected, indicating significant differences in fuel efficiency among the gasoline grades.

Class Time Attendance Comparison via Two-Factor ANOVA

The eighth case examines attendance at three different class times using two-factor ANOVA. It involves calculating sums of squares for treatments, interactions, and error, then constructing the ANOVA table. The critical F-value depends on degrees of freedom. If the F-statistic for treatments exceeds the critical value, it indicates significant differences between class times. Post-hoc tests may identify specific groups differing.

Effects of Lab and Battery Type on Battery Life

The ninth scenario involves a two-factor ANOVA with interaction, testing whether lab and battery type influence battery life. Calculations include sum of squares for factors, interaction, residuals, and mean squares. The F-statistic for interaction tests if the effect of one factor depends on the level of the other. A significant interaction means that the effect of the lab varies by battery type, which impacts interpretation and further analysis.

Regression Analysis and Correlation of Employment Duration and Absence

The tenth analysis develops a regression model predicting the number of workdays absent based on years of employment. The method involves calculating the slope and intercept, standard error, and correlation coefficient, then testing the significance of the correlation. The coefficient of determination indicates the proportion of variance explained. These calculations guide understanding the relationship pattern and its statistical significance.

Scatter Plot and Correlation between Labor Hours and Cost

The final scenario involves plotting a scatter diagram for labor hours applied versus labor cost, identifying the nature of the correlation (positive or negative). The independent variable is hours, and the dependent variable is cost. The pattern of data points reflects the correlation type, which can be confirmed via the correlation coefficient calculation, testing whether it differs significantly from zero using a t-test.

Conclusion

In conclusion, each scenario demonstrates core statistical techniques essential in research and business decision-making. Proper formulation of hypotheses, calculation of test statistics, critical value comparison, and interpretation form the backbone of data-driven insights. Mastery of these methods enables analysts to draw valid conclusions, assess claims accurately, and contribute to evidence-based strategies.

References

• Montgomery, D.C. (2017). Design and Analysis of Experiments. Wiley.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• DeGroot, M.H., & Schervish, M.J. (2012). Probability and Statistics. Addison-Wesley.
• Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
• Casella, G., & Berger, R.L. (2002). Statistical Inference. Duxbury.
• Tabachnick, B.G., & Fidell, L.S. (2013). Using Multivariate Statistics. Pearson.
• Levine, D.M., Stephan, D.F., & Krehbiel, T.C. (2015). Statistics for Managers Using Microsoft Excel. Pearson.
• Field, A. (2013). Discovering Statistics Using SPSS. Sage.
• McClave, J.T., & Sincich, T. (2018). Statistics. Pearson.
• Hogg, R.V., McKean, J., & Craig, A.T. (2014). Introduction to Mathematical Statistics. Pearson.