Social interaction of mental patients. The Community Mental...

Social interaction of mental patients. The Community Mental Health Journal (Aug. 2000) presented the results of a survey of over 6,000 clients of the Department of Mental Health and Addiction Services (DMHAS) in Connecticut. One of the many variables measured for each mental health patient was frequency of social interaction (on a 5 -point scale, where $1=$ very infrequently, $3=$ occasionally, and $5=$ very frequently). The 6,681 clients who were evaluated had a mean social interaction score of 2.95 with a standard deviation of 1.10 . a. Conduct a hypothesis test (at $\alpha=.01$ ) to determine whether the true mean social interaction score of all Connecticut mental health patients differs from 3 . b. Examine the results of the study from a practical view, and then discuss why "statistical significance" does not always imply "practical significance." c. Because the variable of interest is measured on a 5-point scale, it is unlikely that the population of ratings will be normally distributed. Consequently, some analysts may perceive the test from part a to be invalid and search for alternative methods of analysis. Defend or refute this position.

Social interaction of mental patients. The Community Mental Health Journal (Aug. 2000) presented the results of a survey of over 6,000 clients of the Department of Mental Health and Addiction Services (DMHAS) in Connecticut. One of the many variables measured for each mental health patient was frequency of social interaction (on a 5 -point scale, where $1=$ very infrequently, $3=$ occasionally, and $5=$ very frequently). The 6,681 clients who were evaluated had a mean social interaction score of 2.95 with a standard deviation of 1.10 .
a. Conduct a hypothesis test (at $\alpha=.01$ ) to determine whether the true mean social interaction score of all Connecticut mental health patients differs from 3 .
b. Examine the results of the study from a practical view, and then discuss why "statistical significance" does not always imply "practical significance."
c. Because the variable of interest is measured on a 5-point scale, it is unlikely that the population of ratings will be normally distributed. Consequently, some analysts may perceive the test from part a to be invalid and search for alternative methods of analysis. Defend or refute this position.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make a decision about a population parameter based on sample data. It involves setting up a null hypothesis (a statement of no effect or no difference) and an alternative hypothesis (what the researcher aims to support), calculating a test statistic from the data, and then comparing this statistic to a critical value determined by the significance level. This process helps determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

Significance Level and Test Statistic

The significance level, often denoted by alpha (?), defines the threshold for rejecting the null hypothesis and represents the probability of committing a Type I error—incorrectly rejecting a true null hypothesis. The test statistic, which is based on the sample data, quantifies the difference between the observed value and the value posited by the null hypothesis. It is subsequently compared against a critical value or used to derive a p-value that indicates the probability of observing such data if the null were true.

Statistical Significance versus Practical Significance

Statistical significance indicates that an observed effect is unlikely to have occurred by chance, as measured by the p-value or test statistic relative to a pre-set significance level. However, practical significance addresses whether the magnitude of this effect has real-world importance or practical implications. A result can be statistically significant yet of trivial practical importance, particularly in large samples where even tiny effects can achieve statistical significance.

Central Limit Theorem and Normality Assumptions

The Central Limit Theorem (CLT) states that the distribution of sample means will tend to be approximately normal if the sample size is sufficiently large, regardless of the shape of the original population distribution. This principle is crucial when the population data do not follow a normal distribution, as it justifies the use of parametric tests (like t-tests) for hypothesis testing. Although measurements on a bounded scale may not be normally distributed at the individual level, the CLT supports the validity of tests on the mean given a large sample size.

Transcript

00:01 For this problem, we have a bit of background information given, but what we really care about is, oh, actually, no, this is going to be relevant.

00:08 We're told that one of the many variables measured in a survey of mental health patients was frequency of social interaction on a five -point scale, where one is very infrequently, three is occasionally, and five is very frequently.

00:21 We have that the sample size, n, is 6 ,6181, and we have that the mean social interaction score, x bar is 2 .95, with a standard deviation, s, of 1 .1.

00:36 In part a, we are asked to conduct a hypothesis test at alpha equals 0 .01 to determine whether the true mean social interaction score of all connecticut mental health patients differs from three.

00:49 So we have that our null hypothesis here is that mu equals 3, the alternate hypothesis, that mu does not equal 3, and we have that we can use the z statistic because we have a sufficiently large value for n.

01:07 So z will be equal, or the critical value of z we can find from a table, something along those lines.

01:13 We'll find to be 2 .576 or specifically plus or minus 2 .576.

01:19 So we'll be able to reject if the absolute value of our z score is greater than 2 .576.

01:28 Let me fix that there.

01:29 2 .576.

01:32 Calculating the z score, we'll have that it would be x bar, 2 .95, minus mu, 3, divided by s, 1 .1, over the square root of n, so over the square root of 6 ,681...

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