Sampling Variability and Standard Error
Problem 1
People in a large population average 60 inches tall. You will take a random sample and will be given a dollar for every person in your sample who is over 65 inches tall. For illustration if you sample 100 people and 20 turn out to be over 65 inches tall, you get $20. Which is better: a sample of size 100 or a sample of size 1,000? Select one and explain. Does the law of averages relate to the answer you give?
Problem 2
Based on a simple random sample of one hundred an analyst calculate the average hourly wage earned by workers in a city to be $30 and computes the margin of error to be $5. Will we conclude from this that most workers there earn between $25 and $35 per hour? Is this the right interpretation for the margin of error?
Problem 3
Polls showed the two main candidates in the 2004 presidential election were nearly tied on the day before the election. To select the winner a newspaper would lie to have a poll that has a margin of error of less than 1%. Roughly how large a sample would be needed for such a poll?
Confidence Intervals
Problem 1
A large population of overdue bills has balances that follow a normal curve. When we take a sample of 100 of these the average is $500 and the standard deviation is $100. (a) What statement can you make about the range $300 to $700? (b) What statement can you make about the range $480 to $520?
Problem 2
Pollsters try to determine whether or not a person is a "likely voter" before they count their opinion in a poll. If we assume 40% of the registered voters will actually vote, in a random sample of 100 registered voters we can be 95% confident that somewhere between ____________ and _____________ of them will actually vote. Fill in the blanks with a number
Problem 3
An investment firm with 10,000 clients would like to accurately forecast the average dollar amount their current customers will deposit over the coming year. They decide to telephone a random selection of 25 of their customers to ask how much they plan to deposit, and they would like to keep this sample as small as possible so the calls do not annoy too many customers. Since they will be multiplying this average by the total number of customers to get an overall forecast, they would like to accurately estimate this average with a margin of error of less than $4000. Last year the average deposit for all 10,000 clients was $25,000 with a standard deviation of $30,000. Do you think a sample of 25 is enough to give them the margin of error they want? If not, how large a sample do you suggest they need to take? Justify your answer with relevant calculations.
Hypothesis Testing
Problem 1
Over the last year the absentee rate at a large corporation averaged 8.2 days absent with a standard deviation of 6 days. One department with 40 employees had an absentee rate of 12 days per employee. During an investigation the department head argued as follows. "If you took 40 employees at random from the corporation, there is a pretty good chance the average number of days absent would be 12 or more. That's what happened to us-Chance variation. Is this a good defense?
Problem 2
Market researchers could like to know if customers prefer a well-known brand over a generic brand of soft drink. They give a large sample of people the two drinks to taste in a random order and ask them which one tastes better. They find that 70% of people say the branded one and researchers evaluate a p-value of .02. Interpret this number.
Problem 3
A candidate must gather at least 8000 valid signatures on a petition before the deadline in order to run in an election. One candidate turns in 10,000 signatures right before the deadline, but it's expected that some percentage of them are invalid. Election officials take a random sample of 100 signatures and thoroughly investigate them to find that 84% are valid. Is this statistically significant evidence that het candidate has enough valid signatures overall? Explain.