Confidence Intervals with Means

1. Confidence Intervals with Means

2. What is the purpose of a confidence interval? To estimate an unknown population parameter

3. Formula: Standard deviation of statistic Critical value statistic Margin of error

4. In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, taking 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?

5. 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 df = n - 1

6. Graph examples of t- curves vs standard normal curve

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

8. Formula: Standard deviation of statistic Standard error – when you substitute s for s. Critical value statistic Margin of error

9. How to find t* Can also use invT on the calculator! Need upper t* value with 5% is above – so 95% is below invT(p,df) • Use Table B for t distributions • 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* =2.132 t* =2.145

10. Steps for doing a confidence interval: • Assumptions – • Calculate the interval • Write a statement about the interval in the context of the problem.

11. Statement: (memorize!!) We are ________% confident that the true mean context is between ______ and ______.

12. Assumptions for t-inference • Have an SRS from population (or randomly assigned treatments) • s unknown • Normal (or approx. normal) distribution • Given • Large sample size • Check graph of data Use only one of these methods to check normality

13. Ex. 1) Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. • Assumptions: • Have randomly assigned males to treatment • Systolic blood pressure is normally distributed (given). • s is unknown • We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.

14. Ex. 2) 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. We are 95% confident that the true mean pulse rate of adults is between 70.883 & 74.497.

15. Ex 2 continued) 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.

16. Ex. 3) 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. We are 98% confident that the true mean calorie content per serving of vanilla yogurt is between 126.16 calories & 189.56 calories.

17. Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than! Ex 3 continued) A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate? 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.

18. Robust CI & p-values deal with area in the tails – is the area changed greatly when there is skewness • An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the normality assumption is violated. • t-procedures can be used with some skewness, as long as there are no outliers. • Larger n can have more skewness. Since there is more area in the tails in t-distributions, then, if a distribution has some skewness, the tail area is not greatly affected.

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

20. Ex 4) The heights of PWSH male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval? n = 43

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

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