Question

Do You Find Solitude Distressing? "For many people, being left alone with their thoughts is a most undesirable activity," says a psychologist involved in a study examining reactions to solitude. \({ }^{26}\) In the study, 146 college students were asked to hand over their cell phones and sit alone, thinking, for about 10 minutes. Afterward, 76 of the participants rated the experience as unpleasant. Use this information to estimate the proportion of all college students who would find it unpleasant to sit alone with their thoughts. (This reaction is not limited to college students: in a follow-up study involving adults ages 18 to 77 , a similar outcome was reported.) (a) Give notation for the quantity being estimated, and define any parameters used. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated, given that the margin of error for the estimate is \(8 \%\).

Short answer

The best estimate for the population proportion P (the proportion of all college students who find solitude distressing) is 0.52. The 95% confidence interval for this estimation, given a margin of error of 0.08, is (0.44, 0.60).

Step-by-step solution

Notations and Definitions

To begin with, let's define the parameters and the quantity being estimated: \n\n\( P \) - the true population proportion of all college students who find solitude distressing.\n\n\( \hat{P} \) - the sample proportion. \n\nWe are given the number of participants (n = 146) and the number of those participants who found solitude distressing (x = 76). This allows us to calculate \(\hat{P} = x/n \)

Compute Sample Proportion

Next, find the sample proportion. From the data given, we know that 76 out of 146 participants found solitude distressing this can be calculated as \(\hat{P} = 76/146 \approx 0.52\). This is the best estimate for the quantity we're trying to find, P.

Confidence Interval

Finally, with the margin of error (E) provided as 8%, we can construct the confidence interval for the proportion P. A confidence interval is computed as \(\hat{P} \pm E\).\n\nPlug our numbers in and the interval will become \(0.52 \pm 0.08\). This simplifies to the interval (0.44, 0.60).