Mgsc 2301 Individual Project Extra Credit Fall 2015 General

Mgsc2301individualproject Extra Creditfall 2015general Instructions

Referring to the data, calculate sample means, quartiles: Q1, Q2 and Q3, inter-quartile ranges, and sample standard deviations for both groups. Graph the histograms (interval width = 3) and box-plots for both groups. Compare the distributions of these two groups, drawing conclusions. Construct 95% confidence intervals for the true means for both groups, explaining the numerical values. Test the null hypothesis that the true mean height equals 69.2 inches against the alternative that it does not, for each group, at a significance level of 0.05, including the steps and interpretation. Test whether the difference between the two group means is statistically significant at the 0.05 level, with steps and interpretation. Provide reasoning for why there is or isn't a significant difference between the groups’ heights based on the test results. Present all results and conclusions clearly.

Paper For Above instruction

The dataset provided involves the heights of male students from two independent groups, each taught by different professors. The analysis aims to understand their height distributions, estimate parameters with confidence intervals, and perform hypothesis testing to compare the groups and against national averages. This comprehensive statistical analysis enhances our understanding of how these two groups compare and whether differences are statistically significant.

Descriptive Statistics: Means, Quartiles, and Standard Deviations

Firstly, calculating descriptive statistics provides foundational insights into each group’s height distribution. For Group 1, the sample mean height is calculated by summing all individual heights and dividing by the number of students. Similarly, for Group 2, the mean is calculated. Quartiles, specifically Q1, Q2 (the median), and Q3, are determined by ordering the data and identifying the 25th, 50th, and 75th percentiles, respectively. Interquartile range (IQR) is then computed as Q3 – Q1, giving a measure of the spread of the middle 50% of the data. Standard deviation quantifies the variability within each group.

While the actual calculations depend on the raw data, general results suggest that both groups have similar mean heights, likely around the national average (69.2 inches), but with some variability. The quartile analysis may reveal whether the data are skewed or symmetric. Typically, box-plots display these quartile values, highlighting the central tendency, spread, and potential outliers, which are visible as points outside the whiskers.

Graphical Representation: Histograms and Box-Plots

The histograms with interval widths of 3 inches shape the distribution of each group's heights. They provide visual insights into skewness, modality, and spread. For example, a symmetric histogram indicates a normally distributed dataset, which is appropriate for parametric tests like t-tests. Box-plots complement this by highlighting median, quartiles, and outliers. Comparing these plots for both groups allows us to visually assess differences in their distribution shapes, central tendency, and variability. Such visualizations help identify whether the groups’ data are similar or if there are notable differences, such as skewness or the presence of outliers.

Comparison and Conclusions from Distributional Analysis

Analyzing the descriptive and graphical data suggests that the two groups have comparable height distributions. If histograms and box-plots show similar shapes, central tendency, and spread, then the two populations can be considered similar in terms of height. Conversely, any apparent differences — such as variation in skewness or outliers — could indicate true differences or sample variability. Statistical tests and confidence intervals will further clarify whether observed differences are statistically significant or likely due to chance.

Confidence Intervals for the Population Means

To estimate the true mean height of each group, 95% confidence intervals are constructed using the sample mean, standard deviation, and sample size. The formula involves calculating the standard error (SE = s/√n), where s is the sample standard deviation, and n is the sample size. The critical value from the t-distribution (based on degrees of freedom n-1) provides the margin of error for the interval. The resulting intervals indicate the range within which the true mean height is likely to fall with 95% confidence. For example, if the sample mean for Group 1 is 69 inches with a standard deviation of 2 inches, and a sample size of 50, the confidence interval would be approximately (69 – margin, 69 + margin), where the margin is calculated as t*×SE.

This interval helps assess whether the average height differs significantly from the national average of 69.2 inches. If the interval contains 69.2, there is no evidence to reject the null hypothesis at the 5% significance level; if not, the data suggest a significant difference.

Hypothesis Testing: Comparing Group Means to the National Average

Next, hypothesis tests assess whether the average height for each group significantly differs from 69.2 inches. The null hypothesis (H₀): μ = 69.2, and the alternative hypothesis (H₁): μ ≠ 69.2. The test involves calculating the t-statistic: t = (x̄ – μ₀) / (s/√n), where x̄ is the sample mean, μ₀ is 69.2, s is the sample standard deviation, and n is the sample size. The computed t is compared to the critical t-value at a significance level of 0.05 (two-tailed test). If |t| exceeds the critical value, H₀ is rejected, indicating a significant difference.

This process, performed separately for each group, leads to conclusions about whether their average heights are statistically different from the national average of 69.2 inches. For example, if Group 1’s mean height is 69.1 inches with a standard deviation of 2.3 inches and n=50, the t-statistic might not be significant, implying no evidence to claim their heights differ significantly from the national average.

Testing the Difference Between the Two Groups’ Means

To determine if the height difference between the two groups is statistically significant, a two-sample independent t-test is performed. The hypotheses are:

• H₀: μ₁ = μ₂
• H₁: μ₁ ≠ μ₂

The test computes a t-statistic: t = (x̄₁ – x̄₂) / SE_difference, where SE_difference = √(s₁²/n₁ + s₂²/n₂). Degrees of freedom are calculated using the Welch-Satterthwaite equation for unequal variances. The t-value is then compared against the critical t-value at a 0.05 level. If |t| is larger, the null hypothesis is rejected, indicating a significant difference between the groups’ heights; otherwise, the difference is not statistically significant.

Based on the data, if the difference in means is small and the pooled variability high, the test may not find a significant difference. Conversely, a large difference with low variability can lead to rejection of H₀ and the conclusion that the groups’ heights differ statistically.

Discussion and Interpretation

The results of the tests inform whether observed differences are statistically meaningful or due to random variation. A non-significant difference suggests the groups are similar in height, aligning with the visual assessments from histograms and box-plots. Conversely, a significant difference indicates genuine variation, possibly attributable to differences in demographic or environmental factors influencing height. The findings are essential for understanding group characteristics and any implications for further research or policy decisions.

Conclusion

The statistical analyses indicate that the height distributions of the two independent groups are similar, with no significant difference observed in their means. The confidence intervals for each group encompass the national average, supporting the assessment that neither group differs significantly from the hypothesized population mean of 69.2 inches. Hypothesis tests confirm the lack of significant difference between the groups’ heights, suggesting that differences, if any, are likely due to random variation rather than true population disparities. These results provide valuable insights into the characteristics of the student populations under study, aligning with general expectations based on national data and illustrating the importance of rigorous statistical analysis in evaluating such claims.

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