Introduction to Inference Confidence Intervals

The Task Turns out your estranged uncle’s, cousin's, next-door neighbor’s best friend is in the lucrative profession of JellyBlubber breeding. He purchased a colony of 100 JellyBlubbers and has been feeding them according to their strict diet. To find out how much the colony is worth, he needs to know what the average width of his colony.

What to do? Turns out the JellyBlubbers are easy to identify (they’re numbered), but are very difficult to catch. How do we estimate the true average width of the colony? How confident are you in the estimate? –D–Dart analogy. estimate margin of error

Confidence Intervals The confidence interval is constructed so that with a determined degree of confidence, the true value of the characteristic will be found inside the interval. The confidence level associated with an interval is the approximate success rate of the method used to construct the interval. estimate margin of error

Confidence Interval for a Population Mean Draw a random sample of size n from a population having mean  and standard deviation . The confidence interval for estimating  is estimate margin of error

Confidence Interval for a Population Mean Draw a random sample of size n from a population having mean  and standard deviation . The confidence interval for estimating  is z* is the corresponding z-score for a determined confidence level.

Inference procedure overview State the procedure Define any variables Establish the conditions (assumptions) Use the appropriate formula Draw conclusions

Example Oxides of nitrogen (called NOX for short) emitted by cars a trucks are important contributors to air pollution. The amount of NOX emitted by a particular model varies from vehicle to vehicle. For one light truck model, NOX emissions vary with unknown mean  and standard deviation .4 grams per mile. You test an SRS of 50 of these trucks and find the sample mean NOX level = 8 grams per mile. Give a 90% confidence interval for the mean NOX level.

Confidence interval  = true mean NOX level Given random sample. Sample is large (n >40), the Central Limit Theorem guarantees an approximately normal sampling distribution. Safe to infer the population is at least 500 trucks to ensure independence.

Continued… For a 90% confidence level, the corresponding z* is We are 90% confident that the true mean NOX level lies between and grams per mile.

Continued… For a 95% confidence level, the corresponding z* is We are 95% confident that the true mean NOX level lies between and grams per mile

Example The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true proportion mean. To determine the required sample size, we must have a value for . The financial aid office is pretty sure that the amount spent on textbooks varies widely, with most values between $50 and $450. For  = 100, how large a sample should be used to be 95% confident of achieving this level of accuracy?

Continued… Need a sample size of at least 97 textbooks.