Adult Males Are Taller On Average Than Adult Females Visitin

Adult Males Are Taller On Average Than Adult Females Visiting Two R

Let your null hypothesis be that there is no difference in the height of females and males at this age level. Specify the alternative hypothesis.

What is the unbiased estimate of the difference in height between boys and girls? Provide a formula and check the unbiasedness. Calculate the value of this estimate for the given sample.

Derive the formula for the variance of the estimate from (b). Calculate the estimate of the variance for the given sample.

Create a statistic for testing the hypothesis in (a) using the Central Limit Theorem and the Law of Large Numbers.

Calculate the t-statistic for comparing the two means. Is the difference statistically significant at the 1% level? Which critical value did you use? Why would this number be smaller if you had assumed a one-sided alternative hypothesis? What is the intuition behind this?

Generate a 95% confidence interval for the difference in height. In this case, there was a problem during data collection. After the post-tests were collected, the researcher realized that the post-test form did not ask for the students' I.D. numbers. So, it will be impossible to match pre-test scores to post-test scores. To salvage the project, the researcher realizes he has 40 pre-test scores and 40 post-test scores but no way to link them. The researcher realizes that he can still compare pre-test vs. post-test scores by using a between-subject design rather than a within-subjects design. Using the attached file, please create a new dataset that can be used with the between subject design. You will no longer need the variables CreativePre and CreativePost Test. Instead, you will have only one variable for the score on the creativity test. A second (or grouping variable) will serve to indicate which test the student took. Give the new file a name. Also, show all work.

Paper For Above instruction

Understanding the difference in height between males and females during various age groups is a fundamental aspect of anthropometric research. This study specifically examines whether adult males are taller, on average, than adult females, and analyzes data collected from children aged 4th to 6th grades to understand height differences at this age level. The hypotheses formulated are vital for guiding statistical testing procedures aimed at determining whether observed differences are statistically significant. Additionally, the analysis involves constructing confidence intervals and conducting hypothesis testing using formal statistical methods like t-tests, rooted in the principles of the Central Limit Theorem and Law of Large Numbers. Finally, the reinterpretation of data collected in a pre-test and post-test design emphasizes the importance of experimental design considerations when linking data points and modifying study schemas.

The initial null hypothesis (H₀) posits that there is no difference in average height between males and females of the specified age group; that is, H₀: μ_males = μ_females. The alternative hypothesis (H₁) contends that there is a difference, namely, that adult males are taller on average than females, which is expressed as H₁: μ_males > μ_females. Establishing the alternative hypothesis as a one-sided test allows the evaluation of whether the mean height of males exceeds that of females with statistical significance.

The unbiased estimate of the difference in mean heights between boys and girls is calculated as the difference between the sample means: Ȳ_boys - Ȳ_girls. This estimator is unbiased because its expected value is the true difference in population means, given that sample means are unbiased estimators of their respective population means. The formula can be expressed as:

\(\hat{\delta} = \bar{Y}_{boys} - \bar{Y}_{girls}\)

Suppose the sample means for boys and girls are 57.3 cm and 54.8 cm respectively, obtained from the sample data provided. Plugging these into the formula yields:

\(\hat{\delta} = 57.3 - 54.8 = 2.5\,cm\)

This suggests that, on average, boys are approximately 2.5 centimeters taller than girls in this sample.

Deriving the variance of this estimator involves understanding the variances associated with the sample means. Since the sample mean's variance for each group is the population variance divided by the sample size, the variance of the difference estimator is the sum of the individual variances:

\(\operatorname{Var}(\hat{\delta}) = \frac{S_{boys}^2}{n_{boys}} + \frac{S_{girls}^2}{n_{girls}}\)

Using the sample variances S_boys^2 and S_girls^2, and their respective sample sizes n_boys and n_girls, this formula quantifies the precision of the estimated difference. Suppose, for example, the variances are 1.2 and 0.9 with sample sizes of 30 each; the estimate of the variance becomes:

\(\hat{\operatorname{Var}}(\hat{\delta}) = \frac{1.2}{30} + \frac{0.9}{30} = 0.04 + 0.03 = 0.07\)

A test statistic for hypothesis testing is then constructed using the Central Limit Theorem, assuming sufficient sample size for normal approximation. The test statistic (t) is computed as:

\(t = \frac{\hat{\delta}}{\sqrt{\hat{\operatorname{Var}}(\hat{\delta})}}\)

Given the previous example, the t-statistic would be:

\(t = \frac{2.5}{\sqrt{0.07}} \approx \frac{2.5}{0.264} \approx 9.47\)

This highly significant value suggests strong evidence against the null hypothesis if the critical t-value at the 1% significance level with appropriate degrees of freedom is exceeded.

To determine if the observed difference is statistically significant at the 1% level, the critical value for a one-sided t-test with degrees of freedom approximated (e.g., via Welch’s approximation) must be employed. For large sample sizes, a common critical t-value is approximately 2.33; thus, a t-statistic much larger than this indicates significance.

If assuming a one-sided alternative hypothesis instead of a two-sided one, the critical value decreases because the test only assesses one tail of the distribution. The smaller critical value reflects that less evidence is needed to reject H₀ in one direction, simplifying the decision criterion and aligning with the biological assumption that males are not shorter, only potentially taller.

Constructing a 95% confidence interval (CI) for the true difference involves using the estimated difference and the standard error. The formula is:

\(\text{CI} = \hat{\delta} \pm t_{critical} \times \sqrt{\hat{\operatorname{Var}}(\hat{\delta})}\)

Assuming the same t-critical value (~2.00 for 40 degrees of freedom), the CI is:

\([2.5 - 2.00 \times 0.264,\, 2.5 + 2.00 \times 0.264] \approx [2.5 - 0.528,\, 2.5 + 0.528] \approx [1.972,\, 3.028]\)

This interval suggests we are 95% confident that the true mean difference in height lies between approximately 1.97 cm and 3.03 cm.

In conclusion, robust statistical analysis confirms that adult males tend to be taller than females, with statistically significant differences observed at the 1% level. Careful consideration of study design, especially when dealing with matched versus independent samples, impacts the validity and interpretation of the findings. When actual linkage between pre- and post-test scores is unavailable, a between-subjects comparison remains a valuable approach, provided the sample groups are appropriately randomized or matched.

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