Question

A recent study shows that just one session of cognitive behavioral therapy can help people with insomnia.1 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) Fill in the following two-way table for this data.

(b) What proportion of the 40 people in the study reported sleep improvement?
Round your answer three decimal places.

(c) Of the people receiving the therapy session, what proportion reported sleep improvements?
Round your answer three decimal places.

(d) What proportion of people who did not receive therapy reported sleep improvements?
Round your answer three decimal places.

(e) If we use to denote the proportion from part (c) and use to denote the proportion from part (d), calculate the difference in proportion reporting sleep improvements, , between those getting therapy and those not getting therapy.
Round your answer three decimal places."

Answer 1

The proportion of people in the study that reported sleep improvement is 0.425. For those who received therapy, the proportion is 0.700, whereas 0.150 of those who didn't receive therapy reported improvements. The difference in proportion reporting sleep improvements between the groups is 0.550.

Step-by-step explanation:

The study involved two groups of people, one group receiving cognitive behavioral therapy and the other did not. The two-way table using the information from the study would look like this:

Therapy Group (Improved Sleep): 14

Therapy Group (No Improvement): 6

No Therapy Group (Improved Sleep): 3

No Therapy Group (No Improvement): 17

(b) To find the proportion of the 40 people in the study that reported sleep improvement, we calculate 17/40 = 0.425, rounding to three decimal places gives 0.425.

(c) The proportion of people receiving the therapy session that reported sleep improvements is calculated as 14/20 = 0.700, rounded to three decimal places.

(d) The proportion of people who did not receive therapy and reported sleep improvements is 3/20 = 0.150, rounded to three decimal places.

(e) The difference in proportion reporting sleep improvements (πtherapy - πno therapy) is calculated as 0.700 - 0.150 = 0.550, rounded to three decimal places.