Confidence Intervals Chapter 9

How confident are you? Guess my weight… –within 50 pounds –within 20 pounds –within 1 pound Shoot a basketball… –at a swimming pool –at a large trash can –at a carnival hoop

What happens to your confidence as the interval gets smaller? The smaller the interval, the lower your confidence. % % % %

Point Estimate Use a statistic to estimate a parameter Simplest approach, but not always very precise  Lots of variation  Lots of variation in the sampling distribution

Confidence Intervals Use an interval to estimate a parameter Formula: (on green sheet) estimate + margin of error

Margin of Error How accurate we believe the estimate is more preciseSmaller margin of error = more precise estimate of the true parameter Formula: based on confidence level

Confidence Level Success rate of the method used to construct the interval ____% of the time, the intervals we construct this way will contain the true parameter

Represents confidence level z* = upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz* Critical Value: z* ("z-star").05 z*= z*= z*= % 95% 99%

Confidence Interval for a Mean: point estimate critical value standard deviation of the statistic margin of error

What does it mean to be 95% confident? 95% chance that  is contained in the confidence interval The probability that the interval contains  is 95% Constructing intervals with this method will produce intervals that contain  95% of the time

Steps for a confidence interval 1)Conditions Randomization is used SRS from population Randomly assigned treatments Sampling distribution is (approximately) normal Population is normal  samp. dist. is normal Large sample size (n > 30)  samp. dist. is approx. normal Graph is roughly symmetrical  samp. dist. is approx. normal  is known

2) Calculate the interval 3) Write statements about the interval in context: We are _____% confident that the true mean context is between ______ and ______. If we made lots of intervals this way, _____% of them would contain the true mean. Steps for a confidence interval

Conditions: SRS of blood measurements Potassium level is normal (given)  known We are 90% confident that the true mean potassium level is between 3.01 and If we made lots of intervals this way, 90% of them would contain the true mean. 1. a) A test for the level of potassium in blood is not perfectly precise. Repeated measurements for the same person on different days vary normally with  = 0.2. A random sample of three patients has a mean of 3.2. What is a 90% confidence interval for the true mean potassium level?

b) Construct a 95% confidence interval for the true mean potassium level. c) Construct a 99% confidence interval for the true mean potassium level. d) What happens to the interval as the confidence level increases? It gets wider – we need a wider net to catch μ with more confidence

How can we make the margin of error smaller? z* smaller (lower confidence level)  smaller (less variation in the population) n larger (to cut the margin of error in half, n must be ___ times as big) Can we really change this? 4

2. A random sample of 7 high school students yields the following SAT scores: Assuming  = 105, construct and interpret a 95% confidence interval for the true mean SAT score. Conditions: SRS of students Boxplot is roughly symmetrical  sampling dist. is approx. normal  known We are 95% confident that the true mean SAT score for high school students is between and If we made lots of intervals this way, 95% of them would contain the true mean.

3. A random sample of 50 high school students has a mean SAT score of Assuming  = 105, find a 95% confidence interval for the mean SAT score of all high school students. Conditions: SRS of students n > 30  sampling dist. is approx. normal  known We are 95% confident that the true mean SAT score for high school students is between and If we made lots of intervals this way, 95% of them would contain the true mean.

Necessary Sample Size If we want a certain margin of error, we can figure out how big a sample we need: Always round up to the nearest whole number!

4. The heights of GBHS male students are normally distributed with  = 2.5 inches. How large a sample is necessary to estimate the true mean height within 0.75 inches with 95% confidence? n = 43 male students

5. In a randomized experiment on the effects of calcium on blood pressure, researchers divided 64 healthy white males at random into two groups, taking calcium or a placebo. The placebo group had a mean systolic blood pressure of with standard deviation 9.3. To determine the effects of the calcium, the researchers need an estimate of the true mean blood pressure for the placebo group. Can we calculate a z-interval with this data? No –  is unknown!

Student’s t-distribution Very similar to normal distribution  Symmetrical, bell-shaped  Area equals 1 Graph examples of t-curves vs. normal curve: Y1: normalpdf(x), Y2: tpdf(x, 2) Y3: tpdf(x, 5) (use "-0") Window: x = [-4,4], xscl =1; y = [0,.5], yscl =1 Change Y3: tpdf(x, 30)

How does t compare to normal? Shorter & more spread out More area in the tails Based on df (degrees of freedom) = n – 1 As n increases, t-distributions become more like standard normal distribution

How to find t-values Use green sheet t-table Look up confidence level at bottom & df on the sides df = n – 1 Find these t*: 90% confidence when n = 5 95% confidence when n = 15 t* = t* = Can also use invT on the calculator! 5% is above, so 95% is below invT(.95, df)

Formula: point estimate Critical value Standard deviation of statistic Margin of error

Conditions for t-inference SRS from population (or randomly assigned treatments) Normal (or approx. normal) sampling distribution –Given that pop. is normal –Large sample size (n > 30) –Graph of data is roughly symmetrical σ unknown  use t

Back to #5: Find a 95% confidence interval for the true mean blood pressure of the placebo group. Conditions: Randomly assigned treatments Blood pressure is normally distributed (given)  unknown We are 95% confident that the true mean blood pressure is between and If we made lots of intervals this way, 95% of them would contain the true mean.

Robustness A robust inference procedure doesn’t change much if the conditions are violated t-procedures can be used with some skewness, as long as there are no outliers Larger n  can handle more skewness t-distributions already have more area in the tails, so if a distribution has some skewness, the tail area is not greatly affected. CIs deal with area in the tails – the area most affected by skewness

6. A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse of beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rate of adults. We are 95% confident that the true mean pulse rate of adults is between and If we made lots of intervals this way, 95% of them would contain the true mean.

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does our evidence refute this claim? Explain. Since the 95% confidence interval contains 72 bpm, our evidence does not refute this claim.

7. Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. We are 98% confident that the true mean calorie content per serving of vanilla yogurt is between and If we made lots of intervals this way, 98% of them would contain the true mean.

A diet guide claims you get 120 calories from a serving of vanilla yogurt. What does our evidence indicate? Since the 98% confidence interval does not contain 120, we have strong evidence to suggest that the average calories per serving is not 120.

Some Cautions: Formulas only work for SRS No way to correct for bias in data Knowing  is unrealistic in practice