Question

In Exercise 1.23, we learned of a study to determine whether just one session of cognitive behavioral therapy can help people with insomnia. In the study, forty people who had been diagnosed with insomnia were randomly divided into two groups of 20 each. People in one group received a one-hour cognitive behavioral therapy session while those in the other group received no treatment. Three months later, 14 of those in the therapy group reported sleep improvements while only 3 people in the other group reported improvements. (a) Create a two-way table of the data. Include totals across and down. (b) How many of the 40 people in the study reported sleep improvement? (c) Of the people receiving the therapy session, what proportion reported sleep improvements? (d) What proportion of people who did not receive therapy reported sleep improvements? (e) If we use \(\hat{p}_{T}\) to denote the proportion from part (c) and use \(\hat{p}_{N}\) to denote the proportion from part (d), calculate the difference in proportion reporting sleep improvements, \(\hat{p}_{T}-\hat{p}_{N}\) between those getting therapy and those not getting therapy.

Short answer

From the total 40 people in the study, 17 reported sleep improvement. Among people receiving the therapy, 70% reported sleep improvements while only 15% of the people who did not receive therapy reported improvements. The difference in proportion reporting sleep improvements between those getting therapy and those not getting therapy is 0.55 or 55%.

Step-by-step solution

Create a Two-Way Table

Create a two-way table using data provided. By dividing the total number of people who participated in the study (40) into those who received treatment and those who didn't, and then subdividing these into those reported sleep improvement and those didn't. The result would be a 2x2 table.

Calculate the Total Number of Improvements

According to the data, there were 14 people from the treatment group and 3 people from the non-treatment group who reported sleep improvements. Sum these two amounts to calculate the total number of people who reported improvement, which results in \(14 + 3 = 17\)

Calculate Proportion for Treatment Group

To get the proportion of people who reported improvements after treatment (\(\hat{p}_{T}\)), divide the number of improvements in the treatment group by the total number of people in the treatment group. We get \(\hat{p}_{T} = \frac{14}{20} = 0.7\). This means that 70% of the people receiving treatment reported improvement.

Calculate Proportion for Non-Treatment Group

The proportion of people who reported improvements without treatment (\(\hat{p}_{N}\)) is calculated in the same way as Step 3. Divide the number of people who reported improvement (3) by the total number of people in the non-treatment group (20). Thus, \(\hat{p}_{N} = \frac{3}{20} = 0.15\). Therefore, 15% of the people who did not get any treatment reported sleep improvements.

Calculate the Difference in Proportion

You calculate the difference in the proportions by subtracting the smaller proportion (\(\hat{p}_{N}\)) from the larger proportion (\(\hat{p}_{T}\)). So \(\hat{p}_{T}-\hat{p}_{N} = 0.7 - 0.15 = 0.55\).

Key concepts

Randomized Control Trials

In the context of medical research, and specifically the study mentioned in the exercise, a randomized control trial (RCT) is one of the most reliable ways of determining if a treatment is effective. RCTs involve randomly assigning participants to either the treatment group or the control group. This randomization helps ensure that the two groups are comparable and that the outcomes can be attributed to the treatment rather than other factors. The treatment group receives the therapy or intervention being tested – in this case, cognitive behavioral therapy for insomnia – while the control group does not receive the treatment. By comparing outcomes between the groups, researchers can assess the intervention's effectiveness.

Randomization enhances the validity of the study by balancing unknown or unforeseen confounding variables across the groups. When it comes to cognitive behavioral therapy for insomnia, understanding the impact of a single session is crucial, given that chronic insomnia can have a significant negative impact on quality of life. As shown in the study's results, comparing improvements between the groups provides valuable insights into the therapy's effectiveness.

Proportion Calculation

Proportion calculation is a technique used to describe the relative size of a group that shares a particular characteristic, compared to the whole. The term 'proportion' specifically refers to the fraction of the total that possesses this characteristic. To compute a proportion, you divide the number of subjects having the attribute (like sleep improvement in the study) by the total number of subjects. For instance, in the sleep study, to find the proportion of people who reported sleep improvement after receiving therapy, you would divide the number of therapy recipients who improved (14) by the total number of therapy recipients (20), resulting in a proportion of 0.7 or 70%.

Understanding proportions is essential as they give a normalized measure of impact, allowing for comparison even if the total numbers are different. Proportions are commonly expressed as percentages, making it easier for people to grasp the comparative effects of different treatments, such as the improvements in insomnia symptoms following cognitive behavioral therapy.

Two-Way Table

Two-way tables are extremely useful in statistics as they facilitate the organization and comparison of data between two categories. For instance, in the insomnia therapy study, a two-way table allows for a clear visual of the relationship between receiving therapy and reporting sleep improvements. The rows might represent whether a participant received therapy or not, while the columns show whether the participant reported an improvement. Coupling these, you get four categories: therapy with improvement, therapy without improvement, no therapy with improvement, and no therapy without improvement.

Such tables not only display frequency of outcomes in each combination of categories but also enable easy calculation of totals and proportions. For example, by looking at the totals in the two-way table from the therapy study, one can quickly discern the overall rate of improvement among participants, further facilitating the comparison between the treatment's effectiveness versus that of no treatment.