1 Probability and Statistics Confidence Intervals

2 Example 1: Weights of doctors Experimental question: Are practicing doctors setting a good example for their patients in their weights? Experiment: Take a sample of practicing doctors and measure their weights Sample statistic: mean weight for the sample IF weight is normally distributed in doctors with a mean of 150 lbs and standard deviation of 15, how much would you expect the sample average to vary if you could repeat the experiment over and over?

3 doctors’ weights Standard deviation reflects the natural variability of weights in the population mean= 150 lbs; standard deviation = 15 lbs Relative frequency of 1000 observations of weight -1 SD+1 SD -2 SD +2 SD -3 SD +3 SD

doctors’ weights average weight from samples of 2 -1 SD+1 SD -2 SD+2 SD -3 SD+3 SD 4

5 average weight from samples of SD+1 SD -2 SD+2 SD -3 SD+3 SD 5

6 average weight from samples of SD+1 SD-2 SD+2 SD-3 SD+3 SD 6

7 Provides Range of Values Based on Observations from 1 Sample Gives Information about Closeness to Unknown Population Parameter Stated in terms of Probability Never 100% Sure Confidence Interval Estimation

8 A Probability That the Population Parameter Falls Somewhere Within the Interval. Elements of Confidence Interval Estimation Confidence Interval Sample Statistic Confidence Limit (Lower) Confidence Limit (Upper) Population Parameter

9 Parameter = Statistic ± Its Error Confidence Limits for Population Mean Error = Error =

10 Point estimate  (measure of how confident we want to be)  (standard error) Confidence Intervals The value of the statistic in my sample (eg., mean, odds ratio, etc.) From a Z table or a T table, depending on the sampling distribution of the statistic. Standard error of the statistic.

11 Common “Z” levels of confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Z value % 90% 95% 98% 99% 99.8% 99.9%

12 Approximate Confidence Intervals The normal approximation can be used to compute approximate confidence intervals if the sample size is large (n>30) S.E 1.96 S.E 2.57 S.E 90 % Confidence Interval 99 % Confidence Interval 95 % Confidence Interval Margin of error  -1.96SE   +1.96SE Area under the normal curve = 95%

13 Expressions for C.I.’s The 90% C.I. for the population mean: The 95% C.I. for the population mean: The 99% C.I. for the population mean: s is the standard deviation of the n observations. is the sample average of n observations in a simple random sample of size n, where n is large (>30)

14 General remarks on C.I.’s The purpose of a C.I. is to estimate an unknown parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct. The methods used here rely on the assumption that the sample is randomly selected. Any confidence interval has two parts: estimate ± margin of error The confidence level states the probability that the method will give a correct answer, i.e. the confidence interval contains the “true” value of the parameter. The margin of error of a confidence interval decreases as The confidence level decreases The sample size n increases

15 1.Notice the trade off between the margin of error and the confidence level. The greater the confidence you want to place in your prediction, the larger the margin of error is (and hence less informative you have to make your interval). 2.A C.I. gives the range of values for the unknown population average that are plausible, in the light of the observed sample average. The confidence level says how plausible. A C.I. is defined for the population parameter, NOT the sample statistic. To make a margin of error smaller, you can take a larger sample! Remarks:

16 The true meaning of a confidence interval In the following computer simulation: Imagine that the true population mean is 10. A computer took 50 samples of the same size from the same population and calculated the 95% confidence interval for each sample. Here are the results…

17 95% Confidence Intervals

18 3 misses = 6% error rate For a 95% confidence interval, you can be 95% confident that you captured the true population value. 95% Confidence Intervals

19 Confidence Intervals for our weight example… (Sample statistic) ± (measure of how confident we want to be) × (standard error) 95% CI: 160 ± 1.96 × 1.5 = 157–163 lbs 99% CI: 160 ± 2.58 × 1.5 = 156–164 lbs Note how the confidence intervals do not cross the null value of 150!

20 Duality with hypothesis tests. Null value 95% confidence interval Null hypothesis: Average weight is 150 lbs. Alternative hypothesis: Average weight is not 150 lbs. P-value <

21 Null value 99% confidence interval Null hypothesis: Average weight is 150 lbs. Alternative hypothesis: Average weight is not 150 lbs. P-value < Duality with hypothesis tests.

22 Practice problem 1. Waiting times (in hours) at a popular restaurant are believed to be approximately normally distributed with a SD of 1.5 hr. during busy periods. a. A sample of 20 customers revealed a mean waiting time of 1.52 hours. Construct the 95% confidence interval for the estimate of the population mean. b. Suppose that the mean of 1.52 hours had resulted from a sample of 32 customers. Find the 95% confidence interval. c. What effect does larger sample size have on the confidence interval?

23 Common “Z” levels of confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Z value % 90% 95% 98% 99% 99.8% 99.9%

24 Answer (a) a. A sample of 20 customers revealed a mean waiting time of 1.52 hours. Construct the 95% confidence interval for the estimate of the population mean. 95% Confidence Interval is (0.87, 2.17)

25 Answer (b, c) b. Suppose that the mean of 1.52 hours had resulted from a sample of 32 customers. Find the 95% confidence interval. c. What effect does larger sample size have on the confidence interval? Makes the confidence interval narrower (more precision). 95% Confidence Interval is (0.99, 2.05)

Example

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