Confidence Intervals Estimating a Population Mean Want to

Confidence Intervals: Estimating a Population Mean, μ • • • Want to estimate average of San Diego college students Don’t want to sample EVERY STUDENT Instead, I could take a sample (e. g. , this class) & estimate μ Xbar! This is called a “point estimate” • The purpose of a CI is to give an “interval estimate” of the population mean, μ with some degree of certainty e. g. , “We are 95% confident that the mean age of all San Diego college students is between 18 -24 years old”

• • • Confidence Intervals Used when we can only collect a sample from a large population If we had data from the entire population, we could say EXACTLY what the mean value is with 100% confidence A Confidence Interval (CI) has two parts: Size: range of number values (i. e. , 7. 3 -8. 4) Confidence Level (i. e. , 95%) Trade-off between confidence and size of confidence interval If I want to be 100% confident in my estimate, would my range be very small or very large? If I only need to be 10% confident, my range can be small • • •

What do CI’s Mean? • • • What confidence intervals DO NOT mean: • • “There is a 95% chance that my CI includes the true population mean” - NOT TRUE! Either it DOES or it DOESN’T! “I am 95% confident that my CI includes the true population mean” Even better: “If I ran this study 100 times, and made 100 CI’s in this manner, 95 of my 100 CI’s would include the true population mean”

• Confidence Interval Example z test example: Does the mean GPA of students who take statistics differ from the mean GPA of all college students? Construct a 95% confidence interval, given: μ = 2. 7, σ = 0. 6, Xbar = 2. 85, sample size n = 36. σXbar = 0. 6/√ 36 = 0. 1 μ = 2. 7 z* = -1. 96 z* = +1. 96

Confidence Intervals • • • Xbar = 2. 85 z = ± 1. 96 σXbar = 0. 6/√ 36 = 0. 1 –Xbar = 2. 85 ± 1. 96*(0. 1) –Xbar = 2. 85 ± 0. 196 –Xbar = between 2. 654 and 3. 046

• • So What? ? ? “We are 95% confident that the population mean lies between 2. 654 and 3. 046. ” The confidence interval we constructed based on our sample includes the overall population mean: 2. 70 This is another way of saying that this sample and our population mean aren’t significantly different! That is, we shouldn’t reject H 0 2. 70 2. 654 2. 85 σXbar = 0. 1 3. 046

Confidence Interval Formulas z test CI: Single Sample t CI: Dependent t CI: Independent t CI:

Ch. 12 hw • A sample of 25 statistics students eats a mean of 9 candy bars per month; it is known that the population mean for college students is 8, with a standard deviation of 4. 1 a. Construct a 95% confidence interval for the true population mean of statistics students’ candy bar eating. b. Does this CI indicate a significant difference between statistics students and other students (why or why not)? c. Compare & contrast this analysis with a 2 -tailed z-test on the same data. 2. Say a different sample from another college yielded a 95% confidence interval of 8. 5 ± 1. 5. True or false: a. The true population mean is between 7 and 10. b. About 95% of these students sampled scored between 7 and 10. c. There is a 95% chance that this CI contains the true population mean. d. 5% of confidence intervals constructed this way will not contain the true population mean. 3. The college candy salesman wants a more precise confidence interval to better estimate inventory. Name two ways to obtain a more precise CI.