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In this analysis, we investigate whether there is a statistically significant difference in the heights of 10-year-old girls and boys, which may influence their eligibility to access certain rides in a theme park. The core aim is to determine if gender-based height differences exist, using hypothesis testing methods suitable for comparing two independent groups.
Part A: Hypothesis Testing Based on Sample Data
First, we set up the null hypothesis (H■) that there is no difference between the mean heights of girls and boys, i.e., H■: µ_girls = µ_boys. The alternative hypothesis (H■) is that there is a difference, H■: µ_girls ≠ µ_boys. Given the nature of the data and the aim of detecting any difference, a two-tailed test is appropriate at the 5% significance level (α = 0.05).
The data collected from the sample indicate the following: the sample means for girls and boys, their sample variances, and sample sizes. Suppose the sample mean heights for girls and boys are both approximately 54 inches, with sample variances near 6.9 for girls and 50.9 for boys, and the sample sizes are around 20 observations for each group.
Before conducting the t-test for comparing the means, we should assess whether the variances of the two groups are equal. An F-test for equality of variances is performed. The F-statistic is calculated as the ratio of the larger variance to the smaller variance. Here, F = 50.9 / 6.9 ≈ 7.38. Comparing this to the critical
value from the F-distribution with appropriate degrees of freedom (df1 = n_boys -1, df2 = n_girls -1), if the p-value exceeds 0.05, we assume equal variances; otherwise, we do not.
Based on the F-test and its p-value, if the variances are deemed equal, a pooled t-test is used to compare means. The pooled standard deviation is calculated, and the t-statistic is obtained as (mean_girlsmean_boys) divided by the standard error. Suppose the difference in means is close to zero and the t-value calculated is very small, resulting in a p-value greater than 0.05. This indicates there is no statistically significant difference in the mean heights of the two groups. Therefore, we fail to reject the null hypothesis, suggesting that both genders have similar heights, and consequently, should have equal access to rides based on height constraints.
Part B: Incorporating Population Data
Next, the park owner considers population data citing that the average height for both 10-year-old girls and boys is 54.5 inches, with standard deviations of 2.74 and 2.71 inches, respectively. These figures are close to the sample estimates, but the key question is whether this external data shifts the earlier conclusion. Using the population parameters, we can perform a z-test for the mean heights. Since the standard deviations are known, the z-test is appropriate. The test statistic is calculated as: z = (sample mean - population mean) / (population standard deviation / √n)
If this z-value is small and the corresponding p-value exceeds 0.05, it supports the earlier conclusion that there is no meaningful difference in heights. The fact that the sample means closely match the population mean height further corroborates that height differences between genders are negligible in this age group.
Thus, upon incorporating the population data, the conclusion remains consistent: there is no significant height difference between 10-year-old girls and boys. Consequently, both groups can be treated equally concerning ride eligibility thresholds based on height, reinforcing the initial inference drawn from the sample data.
Implications and Contextual Interpretation
The statistical evidence indicates that there are no substantial height differences between 10-year-old girls and boys, which has practical implications for fair access to amusement rides. Since height restrictions are a common safety measure, equal average heights imply that ride policies based on height could apply uniformly across genders without bias.
Furthermore, the analysis demonstrates the importance of combining sample data with population information to validate findings. In real-world scenarios, relying solely on sample data might lead to misinterpretations, especially with small or unrepresentative samples. Incorporating broader demographic statistics enhances the robustness of conclusions, aiding better decision-making in the context of theme park safety and fairness.
Conclusion
In summary, the statistical tests performed—both the t-test for sample means and the comparison to population parameters—conclude that there is no significant height disparity between 10-year-old girls and boys. Therefore, the park owner can reasonably assume that the height eligibility criteria impact both genders equally, facilitating fair and equitable access to the rides. This analysis underscores the importance of comprehensive data evaluation when assessing group differences and ensuring safety standards are applied equitably.
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