Question

The time required for kindergarten children to assemble a specific Lego creation was measured for children who had been instructed for four different lengths of time. Five children were randomly assigned to each instructional group. The length of time (in minutes) to assemble the Lego creation was recorded for each child in the experiment. $$ \begin{array}{rlll} \hline & {\text { Training Period (hours) }} \\ \hline .5 & 1.0 & 1.5 & 2.0 \\ \hline 8 & 9 & 4 & 4 \\ 14 & 7 & 6 & 7 \\ 9 & 5 & 7 & 5 \\ 12 & 8 & 8 & 5 \\ 10 & 9 & 6 & 4 \\ \hline \end{array} $$ Use the Kruskal-Wallis \(H\) -Test to determine whether there is a difference in the distribution of times for the four different lengths of instructional time. Use \(\alpha=01\).

Short answer

Answer: No, there is no significant difference in the distribution of times required to assemble the Lego creation among the four groups at a significance level of 0.01, as the calculated H value (9.81) was less than the critical chi-squared value (11.3449).

Step-by-step solution

Compute the combined ranks for the data

First, we must rank all of the times, and then compute the combined ranks for each group. $$ \begin{array}{c||c|c|c|c} \hline \text{Training Period} & 0.5 & 1.0 & 1.5 & 2.0 \\ \hline\hline \text{Ranks} & 17 & 12 & 7.5 & 7.5 \\ \hline & 20.5 & 11 & 9.5 & 9.5 \\ \hline & 13 & 4 & 9.5 & 9.5 \\ \hline & 18 & 12 & 13 & 9.5 \\ \hline & 15 & 14 & 9.5 & 7.5 \\ \hline \end{array} $$

Calculate the sums of ranks for each group

Next, we will calculate the sums of ranks for each group and find the average rank for each group. $$ \begin{array}{c|rrrr|r} \hline \text{Group} & R_i & & & & \sum R_i \\ \hline 0.5 & 17 & 20.5 & 13 & 18 & 15 & 83.5\\ \hline 1.0 & 12 & 11 & 4 & 12 & 14 & 53\\ \hline 1.5 & 7.5 & 7.5 & 9.5 & 9.5 & 13 & 47.5\\ \hline 2.0 & 7.5 & 7.5 & 9.5 & 9.5 & 7.5 & 41.5\\ \hline \end{array} $$

Calculate the Kruskal-Wallis H statistic

Now, we will calculate the Kruskal-Wallis H statistic using the formula: $$ H = \frac{12}{N(N+1)}\sum_{i=1}^{k}\frac{T_{i}^{2}}{n_{i}} - 3(N+1) $$ Where: - \(k\) is the number of groups - \(T_i\) is the sum of the ranks of the \(i\)-th group - \(n_i\) is the number of observations in the \(i\)-th group - \(N\) is the total number of observations $$ H = \frac{12}{20(21)} \left[ \frac{83.5^2}{5} + \frac{53^2}{5} + \frac{47.5^2}{5} + \frac{41.5^2}{5} \right] - 3(21) $$ After performing the calculations, we get: $$ H = 9.81 $$

Find the critical chi-squared value

Following, we will find the critical chi-squared value, \(\chi^{2}_{0.01, k-1}\), to compare with the H statistic calculated above. Since the significance level, \(\alpha = 0.01\), and there are 4 groups (\(k = 4\)), the degrees of freedom will be \(k - 1 = 3\). Referring to a chi-square distribution table, we find that: $$ \chi^{2}_{0.01, 3} = 11.3449 $$

Compare the H statistic and the critical chi-squared value

Lastly, we will compare the Kruskal-Wallis H statistic (\(H = 9.81\)) to the critical chi-squared value (\(\chi^{2}_{0.01, 3} = 11.3449\)). Since H < \(\chi^{2}_{0.01, 3}\), $$ 9.81 < 11.3449 $$ we fail to reject the null hypothesis. Thus, at a significance level of 0.01, we cannot conclude that there is a difference in the distribution of times for the four different lengths of instructional time for kindergarten children assembling the Lego creation.

Key concepts

Nonparametric Statistics

When examining data, statisticians often have to decide whether to use parametric or nonparametric statistical methods. Nonparametric statistics come into play when the data does not adhere to the common assumptions associated with parametric tests, such as being normally distributed. Importantly, nonparametric tests do not rely on data coming from distributions with specific parameters.

This approach is particularly useful when you have ordinal data or when the sample size is small, making it hard to satisfy the requirements of parametric tests. Nonparametric methods, such as the Kruskal-Wallis H test, are less powerful than their parametric counterparts but have the significant advantage of requiring fewer assumptions about the data distribution. This makes the Kruskal-Wallis H test an attractive option when analyzing the distribution of times across different groups with varied sample sizes or non-normal distributions, as with our kindergarten Lego assembly task.

Statistical Significance

Statistical significance is a concept that enables researchers to determine whether the results of an experiment or study could be due to chance or whether they reflect a real effect. It is commonly associated with a p-value, which measures the probability that the observed data would occur by random chance if the null hypothesis were true.

In the context of the Kruskal-Wallis H test, statistical significance helps us decide whether any observed differences in assembly times are likely due to the varying training periods, or simply due to random variation. By setting a significance level (denoted by \( \alpha \)), researchers can control the rate of Type I errors – falsely claiming a result is significant when it's not. Common choices for \( \alpha \) are 0.05 or 0.01. If the computed H statistic is greater than the critical value from the chi-square distribution for the chosen \( \alpha \) level, it suggests that at least one of the groups differs significantly. However, if the H statistic is lower, as in our example with \( H = 9.81 \), we fail to reject the null hypothesis, implying that any observed differences could well be due to chance.

Chi-square Distribution

The chi-square distribution is an essential tool in nonparametric statistics, particularly when evaluating statistical significance. It's a family of distributions that take only positive values and are skewed to the right, with the skewness decreasing as the degrees of freedom increase.

In the Kruskal-Wallis H test, once the H statistic is calculated, it needs to be compared against a critical value from the chi-square distribution with the appropriate degrees of freedom, usually \( k - 1 \) where \( k \) is the number of groups being compared. The degrees of freedom partly reflect how 'flexible' the statistical model is. The chi-square distribution tables, or computational tools, provide these critical values, which determine the threshold for rejecting the null hypothesis. In the case of our Lego assembly study, we use the \( \chi^2 \) distribution with three degrees of freedom (since we have four groups) at an alpha level of 0.01 to obtain a critical value which is compared to our computed H statistic of 9.81 to determine statistical significance.