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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 \(8 \%\).

Step by step solution

01

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 \(\hat{P} = x/n \)

02

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.

03

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

In the world of statistics, the confidence interval is an essential concept often used to estimate an unknown population parameter. The confidence interval provides a range of values that is likely to contain the population proportion.
Think of it as a safety net that helps us make more informed guesses about the data. In our case, we're trying to find out how many college students find solitude distressing. We have one sample with some data, but we want to know what this could mean for all students.

• The confidence level, here 95%, tells us how certain we can be about our interval. A higher confidence level means a wider interval.
• The margin of error is what tells us how far our sample results might be from the actual population values we are estimating.
• In our exercise, we calculated that with a margin of error of 8%, the confidence interval for the population proportion is between 0.44 and 0.60.

Therefore, we have a fairly high chance — 95% certainty — that the actual proportion of all students who find such solitude unpleasant falls within this range.

Sample Proportion

The sample proportion is a crucial statistical measure that gives us an estimate of the population proportion from our sample data. In statistics, it is often denoted as \( \hat{P} \), and it represents the fraction of the sample with a specific characteristic.
In our solitude example, the sample proportion tells us about students who found being alone unpleasant.

• To calculate it, we take the number of favorable outcomes (students who dislike solitude) and divide by the total sample size.
• Here, it's calculated using 76 students who found it unpleasant out of 146 total participants, and \( \hat{P} = \frac{76}{146} \approx 0.52 \).
• This formulation helps us estimate the best guess for the actual population proportion based on our sample data.

So, the sample proportion of 0.52 tells us that, in our sample, about 52% of students felt this way about solitude.

Population Proportion

The population proportion is what we aim to estimate when we conduct a survey or study on a sample of the population. Denoted by \( P \), it represents the fraction of the entire population that exhibits a particular characteristic.
Understanding the population proportion is crucial because it gives insights into the behavior or traits within the broader group.

• In our solitude scenario, the population proportion refers to the percentage of all college students who find solitude distressing.
• Unlike the sample proportion, which comes from a specific group we've surveyed, the population proportion is for the entire student population, which we infer from our sample.
• The confidence interval derived from the sample data serves as a helpful tool to estimate where the population proportion may lie.

Thus, our goal is to ensure our sample gives a reliable estimate that can predict this broader behavior accurately within the margin of error specified.