Confidence Intervals with Means

1. Confidence Intervals with Means Chapter 9

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

3. One-Sample z Confidence Interval for m 2. The sample sizen is large (generally n30), and 3. s , the population standard deviation, is known then the general formula for a confidence interval for a population mean m is given by

4. Formula: Standard deviation of parameter Critical value statistic Margin of error

5. Example A certain filling machine has a true population standard deviation  = 0.228 ounces when used to fill catsup bottles. A random sample of 36 “6 ounce” bottles of catsup was selected from the output from this machine and the sample mean was 6.018 ounces. Find a 90% confidence interval estimate for the true mean fills of catsup from this machine.

6. The z critical value is 1.645 90% Confidence Interval (5.955, 6.081) Conclusion: We are 90% confident that the true mean fills of catsup from the machine is between 5.955 oz. and 6.081 oz.

7. In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups. The participants either take calcium or a 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? We only know sample statistics! We do not know population standard deviation!

8. William S. Gossett Quality control engineer for Guiness Brewery in Dublin, Ireland Checked the stout’s quality by performing hypothesis tests Figured out a new family of models Student’s t distributions

9. 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

10. t Distributions

11. How does the t-distributions compare to the standard normal distribution? • Bell-shaped and centered at 0 • Shorter & more spread out • More area under the tails • As n increases, t-distributions become more like a standard normal distribution

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

13. t Distributions Since each t distribution would require a table similar to the standard normal table, we usually only create a table of critical values for the t distributions. Appendix Table 3 in BOB

15. 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

16. Let’s do some comparing: Finding probabilities Normal model: Student’s t models: Normalcdf(1.645, ∞) .04998 tcdf(1.645, ∞, 4) .08766 tcdf(1.645, ∞, 9) .06719 Student’s t models resemble normal models as sample size gets bigger…. tcdf(1.645, ∞, 14) .06111 .05538 tcdf(1.645, ∞, 29) .05157 tcdf(1.645, ∞, 99)

17. Finding critical values: Student’s t models: Normal model: t critical values: z critical values: 95% invT(.975, 4) invNorm(.975) T critical values approach z critical values as sample size increases…. 1.96 2.776 invT(.975, 9) 2.262 invT(.975, 14) 2.145 2.045 invT(.975, 29) 1.984 invT(.975, 99)

18. Steps for doing a confidence interval: • Identify by name or formula One-sample confidence interval for means • Assumptions • Calculate the interval • Write a statement about the interval in the context of the problem.

19. 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

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

21. 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.

22. 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.

23. One-sample confidence interval for means • Assumptions: • random sample of adults • Pulse rate is normally distributed (given). • s is unknown We are 95% confident that the true mean pulse rate of adults is between 70.883 & 74.497.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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!

29. Ex 4) The heights of SHS 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

30. 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

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

32. Confidence Interval Example Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are 82 66 90 84 75 88 80 94 100 91 Find a 90% confidence interval estimate for the true mean number of hours of television watched per week by shut-ins.

33. Calculating the sample mean and standard deviation we have n = 10, = 85, s = 9.843 Can we meet the assumptions for a t –confidence interval? t-critical value of 1.833 by looking on the t table at 90% confidence with df = 9 or invt(.95, 9). (79.295, 90.705) We are 90% confident that the true mean number of hours of television watched per week is between 79.295 hours and 90.705 hours.