Estimating with confidence

How long does a new model of laptop battery last?

How much does the weight of a quarter-pound hamburger at a fast-food restaurant vary after cooking?

How many students in High School North have iPhones?

All the confidence intervals we will meet have a form similar to this:

estimate ± margin of error

Point estimator and point estimate

How long does a new model of laptop battery last?

Here’s a SRS of 30 batteries:

Point estimator:

Point estimate:

x

7.16x

What proportion p of U.S. high school students smoke? The 2007 Youth Risk Behavioral Survey questioned a random sample of 14,041 students in grades 9 to 12. Of these, 2808 said they had smoked cigarettes at least one day in the past month.

Point estimator:

Point estimate:

sample proportion p̂

20.0041,14

2808ˆ p

How much does a Subway footlong vary from a foot?

“Suits filed in Chicago, New Jersey allege sandwich chain is shorting customers on length of their footlong subs” (Chicago Tribune)

Point estimator:

Point estimate:

2xs

2xsThe value of

The sample variance

One plaintiff, used-car dealer Jason Leslie, 32, of Marlboro NJ, reckons he has eaten about 50 meatball Footlongs a year since he was 18 , reports the New York Post. “They advertise in all these commercials, ‘Footlong, Footlong, Footlong,’ and now I feel like an idiot,” he told The Post. “I can’t believe I fell for that trick. The sandwiches are anywhere between a half-inch to an inch shorter . . . I feel cheated.” (Forbes Magazine)

In each of the following settings, determine the point estimator you would use and calculate the value of the point estimate.

The makers of a new golf ball want to estimate the median distance the new balls will travel when hit by mechanical driver. They select a random sample of 10 balls and measure the distance each ball travels after being hit by the mechanical driver. Here are the distances (in yards):

285 286 284 285 282 284 287 290 288 285

The golf ball manufacturer would also like to investigate the variability of the distance travelled by the golf balls by estimating the interquartile range.

The math department wants to know what proportion of its students own a graphing calculator, so they take a random sample of 100 students and find that 28 own a graphing calculator.

http://bcs.whfreeman.com/tps4e/

Confidence level: To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.”

Confidence interval: To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from_____ to _____ captures the actual value of the [population parameter in context].”

Do You Use Twitter?In late 2009, the Pew Internet and American Life Project asked a random sample of 2253 U.S. adults, “Do you ever…use Twitter or another service to share updates about yourself or to see updates about others?” Of the sample, 19% said “Yes.” According to Pew, the resulting 95% confidence interval is (0.167, 0.213).

Interpret the confidence interval and the confidence level

Confidence interval: We are 95% confident that the interval from 0.167 to 0.213 contains the actual proportion p of U.S. adults who use Twitter or another service for updates.Confidence level: In 95% of all possible samples of 2253 U.S. adults,resulting confidence interval would capture the actual population proportion of U.S. adults who use Twitter or another service for updates

AP EXAM TIP On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you understand the difference: the confidence level describes the long-run capture rate of the method, and the confidence interval gives a set of plausible values for the parameter.

Presidential Approval RatingsAccording to gallop.com, on August 13, 2010, the 95% confidence interval for the true proportion of Americans who approved of the job Barack Obama was doing as president was 0.44 0.03.Interpret the confidence interval and the confidence level.

Interval: We are 95% confident that the interval from 0.41 to 0.47 captures the true proportion of Americans who approve of the job Barack Obama was doing as president at the time of the poll.

Level: In 95% of all possible samples of the same size, the resulting confidence interval would capture the true proportion of Americans who approve of the job Barack Obama was doing as president.

How much does the fat content of Brand X hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard deviation σ is 2.84 to 7.55.

Interpret the confidence interval.

We are 95% confident that the interval from 2.84 to 7.55 g captures the true standard deviation of the fat content of Brand X hot dogs.

Interpret the confidence level.

In 95% of all possible samples of 10 Brand X hot dogs, the resulting confidence interval would capture the true standard deviation.

True or false: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation σ. Justify your answer

False. The probability is either 1 (if the interval contains the true standard deviation) or 0 (if it doesn’t).

Exercises on page 481,# 5-13 odd, 17, 19–24 all

Read section 8.2