Researchers wanted to determine if the psychological profile...

Researchers wanted to determine if the psychological profile of healthy children was different than for children suffering from recurrent abdominal pain (RAP) or recurring headaches. A total of 210 children and adolescents were studied and their psychological profiles were graded according to the Child Behavior Checklist $4-18$ (CBCL). Children were stratified in two age groups: 4 to 11 years and 12 to 18 years. The results of the study are summarized in the following table: $$\begin{array}{lccc} & n & \text { Sample Mean } & \text { Sample Variance } \\\hline \text { Control group } & 70 & 11.7 & 21.6 \\\hline \text { RAP } & 70 & 9.0 & 13.0 \\\hline \text { Headache } & 70 & 12.4 & 8.4\end{array}$$ (a) Compute the sample standard deviations for each group. (b) What sampling method was used for each treatment group? Why? (c) Use a two sample $t$ -test for independent samples to determine if there is a significant difference in mean $\mathrm{CBCL}$ scores between the control group and the RAP group (assume that both samples are simple random samples). (d) Is it necessary to check the normality assumption to answer part (c)? Explain. (e) Use the one-way ANOVA procedure with $\alpha=0.05$ to determine if the mean CBCL scores are different for the three treatment groups. (f) Based on your results from parts (c) and (e), can you determine if there is a significant difference between the mean scores of the RAP group and the headache group? Explain.

Researchers wanted to determine if the psychological profile of healthy children was different than for children suffering from recurrent abdominal pain (RAP) or recurring headaches. A total of 210 children and adolescents were studied and their psychological profiles were graded according to the Child Behavior Checklist $4-18$ (CBCL). Children were stratified in two age groups: 4 to 11 years and 12 to 18 years. The results of the study are summarized in the following table:
$$\begin{array}{lccc} & n & \text { Sample Mean } & \text { Sample Variance } \\\hline \text { Control group } & 70 & 11.7 & 21.6 \\\hline \text { RAP } & 70 & 9.0 & 13.0 \\\hline \text { Headache } & 70 & 12.4 & 8.4\end{array}$$
(a) Compute the sample standard deviations for each group.
(b) What sampling method was used for each treatment group? Why?
(c) Use a two sample $t$ -test for independent samples to determine if there is a significant difference in mean $\mathrm{CBCL}$ scores between the control group and the RAP group (assume that both samples are simple random samples).
(d) Is it necessary to check the normality assumption to answer part (c)? Explain.
(e) Use the one-way ANOVA procedure with $\alpha=0.05$ to determine if the mean CBCL scores are different for the three treatment groups.
(f) Based on your results from parts (c) and (e), can you determine if there is a significant difference between the mean scores of the RAP group and the headache group? Explain.

Key Concepts

One-Way ANOVA

One-way ANOVA (Analysis of Variance) is a statistical procedure that is used to compare the means of three or more independent groups to determine if at least one group mean is significantly different from the others. It analyzes the variance within groups compared to the variance between groups and is based on assumptions including normality, homogeneity of variances, and independent observations.

Post Hoc Comparisons

Post hoc comparisons are additional analyses conducted after an ANOVA indicates a significant difference among group means. These tests (such as Tukey's HSD, Bonferroni correction, etc.) help determine exactly which groups differ from each other. Without the specific post hoc comparison between the RAP and headache groups, one cannot conclusively state that the difference between them is significant even if the overall ANOVA is significant.

Normality Assumption

The normality assumption involves the requirement that the data in each group should be approximately normally distributed. This assumption is important for the validity of many statistical tests, such as the t-test, particularly when sample sizes are small. However, for larger samples the central limit theorem often allows for some deviation from normality without significantly affecting the results.

Sampling Methods

Sampling methods refer to the strategies used to select subjects from a larger population. In many studies, different groups may be formed through approaches such as simple random sampling, stratified sampling (which involves dividing the population into distinct subgroups or strata), or convenience sampling. Each method has implications for the generalizability of the results and potential biases in the sample.

Sample Variance

Sample variance quantifies the average squared deviation of each data point from the sample mean. It is the basis for computing the sample standard deviation and is a key descriptive statistic that helps in understanding the dispersion within a dataset.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It is calculated as the square root of the variance and provides insight into how much individual observations deviate from the group mean. In statistical reporting, it helps summarize the spread of the data in the same units as the original measurements.

Two-Sample t-Test for Independent Samples

The two-sample t-test for independent samples is an inferential statistical method used to compare the means of two independent groups to determine whether there is a statistically significant difference between them. This test assumes that the samples are drawn randomly and independently from normally distributed populations with equal variances (although some versions can accommodate unequal variances).

Transcript

00:01 So we're given 70 people in a control group, 70 people that have some abdominal issues, and 70 people with headaches.

00:09 And we're looking at some data that's given.

00:13 And we have our sample standard deviations, just being the square root of our variance here.

00:19 And we had stratified groups, stratified samples.

00:22 They had two age groups in which they took people from each of these categories.

00:26 Now we're asked to perform a two sample t test to see if these means, between the control and this abdominal group, this rap group, are significantly different.

00:37 So i utilized my software, put those values in, there ends up being 130 degrees of freedom.

00:44 I did not pool, and we get a large test statistic with a very small p value.

00:50 Therefore, we do have evidence to reject the nulls, so we can say there is sufficient evidence to say that there is a difference between these two means.

00:58 Now they asked, do we need to know that these are normal distributions? and no, we don't, because the central limit theorem tells us that this sample sizes of 70 are sufficiently large.

01:10 Now, in the next part, we're asked to perform in a nova from that data.

01:15 So i found the grand mean, first of all, to be 11 .03 repeating.

01:21 And then i found the sst by taking 70 times the difference between each of the number...

Please give Ace some feedback