One-Sample Confidence Intervals

Estimate the percentage of the seniors at your school who regularly follow the daily news, the percentage who can name the two US senators from their home state, the percentage who are registered to vote, and the percentage who would almost certainly vote if a presidential election were held today.

Among seniors at your school who are looking for jobs, estimate the average annual salary they expect to earn their first year. Do not include moving allowances or other one-time benefits.

A study by Camilla Benbow and Julian Stanley of children younger than 14 who scored 700 or higher on the math SAT found that $20 \%$ were left-handed, as compared to $8 \%$ of the entire population. Estimate the fraction of students at your school who are left-handed.

A study by Camilla Benbow and Julian Stanley of children younger than 14 who scored 700 or higher on the math SAT found that $20 \%$ were left handed, as compared to $8 \%$ of the entire population. Estimate the fraction of faculty at your school who are left handed.

Ask at least 50 randomly selected US students whether they expect to receive more or less money from Social Security than their parents will receive. Also ask them how much monthly benefits they expect to receive (in today's dollars). Discarding those who say they will receive the same amount as their parents, estimate the overall percentage of students who believe they will receive more than their parents. Also estimate the average monthly benefit that students expect to receive.

College students are said to experience the Frosh 15-an average weight gain of 15 pounds during their first year at college. Test this folklore by asking at least 100 randomly selected second-year students how much weight they gained or lost during their first year at college. Estimate a $95 \%$ confidence interval for the population mean.

People experiencing an earthquake often grossly overestimate how long the quake lasted; for example reporting that a 6 -second quake lasted 30 seconds. Show a random sample of students a memorable event, such a snippet of loud music or you dancing, and then ask them how long this event lasted. Do your data indicate that people are more likely to underestimate or overestimate how long the event lasted?

Persi Diaconis, a famous mathematician, claims that a spun US penny will land tails about $80 \%$ of the time. Put a clean Lincoln penny on a flat surface and hold the coin on its edge with one finger. Then flick the coin with a finger on your other hand. Repeat this experiment 100 times and estimate a $95 \%$ confidence interval for the probability that a spun penny will land tails.

Tell someone that you randomly selected a MLB player who happened to have gotten 174 base hits in 612 times at bat, a .284 batting average. Ask them to estimate a range for the probability that this player will get a base hit in his next time at bat, such that they are $95 \%$ sure that the true probability of this player getting a base hit is inside this range. Compare this to the actual $95 \%$ confidence interval, assuming that batting can be described by the binomial distribution.

Assemble a very large number of marbles, slips of paper, or other objects that are of two different colors or other features; for example, a very large number of marbles, of which $70 \%$ are red and $30 \%$ are blue. Place the marbles inside a closed bag or box and allow someone to reach into the bag or box and withdraw ten marbles and then, based on their results, estimate a range for the fraction of all marbles that are red (or blue), such that they are $95 \%$ sure that the true percentage is inside this range. Use this calculator https://statpages.info/confint.html to determine the exact $95 \%$ confidence interval. Repeat this experiment at least 40 times and see how often the ranges by the participants include the true percentage.

Show someone a large transparent jar of pebbles, jelly beans, or other small objects and ask them to estimate a range for the number of objects in the jar, such that they are $95 \%$ sure that the actual number of objects is in this range. Repeat this experiment at least 40 times. How often did the specified ranges include the actual number of objects?

Show someone a large transparent jar of pebbles, jelly beans, or other small objects and ask them to estimate the number of objects in the jar. Repeat this experiment at least 50 times. Does a $95 \%$ confidence interval include the actual number of objects? How many individuals made guesses that were more accurate than the average of all the guesses?

Have ten or more friends stand in a line outside a closed door, with you the last person in line. Tell your friends standing in line that if anyone asks why they are waiting in line, they should say, "I don't know. I just saw this line and figured it must be something good." Estimate the probability that a person given this answer will join the line.