Confidence Intervals For a Sample Mean Point Estimate

Confidence Intervals For a Sample Mean

Point Estimate • Use a single statistic based on sample data to estimate a population parameter • Simplest approach • But not always very precise due to variation in the sampling distribution

Confidence intervals • Are used to estimate the unknown population mean • Formula: estimate + margin of error

Margin of error • Shows how accurate we believe our estimate is • The smaller the margin of error, the more precise our estimate of the true parameter • Formula:

Confidence level • Is the success rate of the method used to construct the interval • Using this method, ____% of the time the intervals constructed will contain the true population parameter

What does it mean to be 95% confident? • 95% chance that m is contained in the confidence interval • The probability that the interval contains m is 95% • The method used to construct the interval will produce intervals that contain μ 95% of the time.

Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level 90% 95% 99% z*=1. 645 tail area z*=1. 96 z*=2. 576 z*. 05. 025. 005 1. 645. 025 1. 96. 005 2. 576

Confidence interval for a population mean: Standard Critical value deviation of the statistic estimate Margin of error

Steps for doing a confidence interval: 1) Assumptions – • • SRS from population Sampling distribution is normal (or approximately normal) • • Given (normal) Large sample size (approximately normal) Graph data (approximately normal) σ is known 2) Calculate the interval 3) Write a statement about the interval in the context of the problem.

Statement: (memorize!!) We are ____% confident that the true mean context lies within the interval ______ and ______.

A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with σ = 0. 2. A random sample of three has a mean of 3. 2. What is a 90% confidence interval for the mean potassium level? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) σ known We are 90% confident that the true mean potassium level is between 3. 01 and 3. 39.

95% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) σ known We are 95% confident that the true mean potassium level is between 2. 97 and 3. 43.

99% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) σ known We are 99% confident that the true mean potassium level is between 2. 90 and 3. 50.

What happens to the interval as the confidence level increases? the interval gets wider as the confidence level increases

How can you make the margin of error smaller? • z* smaller (lower confidence level) • σ smaller (less variation in the population) • n larger Really cannot (to cut the margin of error in half, n must change! be 4 times as big)

A random sample of 50 WHS students was taken and their mean SAT score was 1250. (Assume σ = 105) What is a 95% confidence interval for the mean SAT scores of WHS students? We are 95% confident that the true mean SAT score for WHS students is between 1220. 9 and 1279. 1

Suppose that we have this random sample of SAT scores: 950 1130 1260 1090 1310 1420 1190 What is a 95% confidence interval for the true mean SAT score? (Assume σ = 105) We are 95% confident that the true mean SAT score for WHS students is between 1115. 1 and 1270. 6.

Find a sample size: • If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Always round up to the nearest person!

The heights of WHS male students is normally distributed with σ = 2. 5 inches. How large a sample is necessary to be accurate within +. 75 inches with a 95% confidence interval? n = 43

In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114. 9 with standard deviation of 9. 3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not?

Student’s t- distribution • Developed by William Gosset • Continuous distribution • Unimodal, symmetrical, bell-shaped density curve • Above the horizontal axis • Area under the curve equals 1 • Based on degrees of freedom

Graph examples of t- curves vs normal curve Normal: T-Curve: 5 dfs T-Curve: 2 dfs T-Curve: 30 dfs

How does t compare to normal? • Shorter & more spread out • More area under the tails • As n increases, t-distributions become more like a standard normal distribution

How to find t* Can also use inv. T on the calculator! • Use Table B for t distributions t* value with 5% at is above • Need Lookupper up confidence level bottom– & df so 95% is below on the sides • df = n – 1 inv. T(p, df) Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2. 132 t* =2. 145

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

Assumptions for t-interval • Have an SRS from population • σ unknown • Normal distribution – Given – Large sample size – Check graph of data

For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. Assumptions: • Have an SRS of healthy, white males • Systolic blood pressure is normally distributed (given). • σ is unknown We are 95% confident that the true mean systolic blood pressure is between 111. 22 and 118. 58.

Robust • An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated. • t-procedures can be used with some skewness, as long as there are no outliers. • Larger n can have more skewness.

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72. 69 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 rates of adults. (70. 883, 74. 497)

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain. The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: 160 200 220 230 120 180 140 130 170 190 80 120 100 170 Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. (126. 16, 189. 56)

A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. Note: confidence intervals tell us What does this evidence indicate? if something is NOT EQUAL – never less or greater than! Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.

Some Cautions: • The data MUST be a SRS from the population • The formula is not correct for more complex sampling designs, i. e. , stratified, etc. • No way to correct for bias in data

Cautions continued: • Outliers can have a large effect on confidence interval • Must know σ to do a z-interval – which is unrealistic in practice