Confidence Intervals for Population Mean

Transcription

1 Confidence Intervals for Population Mean Reading: Sections 7.1, 7.2, 7.3 Learning Objectives: Students should be able to: Understand the meaning and purpose of confidence intervals Calculate a confidence interval for the mean of a Normally distributed population with or without known variance Calculate a confidence interval for the mean of any distribution with a large sample Correctly interpret confidence intervals Determine the sample size needed to be within a given margin of error 1

2 Point Estimation is Not Enough GOAL: Use a sample to make inference about the population parameters. SO FAR: We used point estimation to make a guess about a population parameter. We saw that these guesses (statistics) have distributions. We know that we only see one value not the entire distribution. We do not know how good our guess is. NEXT: Instead of making a single guess at the parameter, we want to determine an interval of values in which the parameter lay with good confidence. 2

3 Finding CIs for a population Mean Three Settings Normal distribution with known variance (contrive but easiest to understand) Normal distribution with unknown variance Note: The above two settings do not require large sample size Non-normal distribution, but sample size is large enough so that the CLT applies. 3

4 Example 1: Normal Distribution Variance Known Suppose the following beam strengths (a subset from a previous example) follow a normal distribution N(µ, ): 5.9, 7.2, 7.3, 6.3, 8.1, 6.8, 7.0, 7.6, 6.8, 6.5, 7.0, 6.3, 7.9, 9.0, 8.2, 8.7, 7.8, 9.7, 7.4, 7.7, 9.7, 7.8, 7.7, 10.7,

5 Example 1 (cont d): CI for μ Point estimator follows a normal distribution x 25 5

6 Interpreting a CI The probability that the random interval contains μ is 1-α. If we have N samples and we compute the CI N times. Then 90% of them will contain the true value of μ. 6

7 Example 1 (cont d): CI for μ for different α levels 7 212Xz1. Xz. 225, 2 / / 25 X2/ α = 0.1; 90% CI = zz/ 2/ 2α = 0.05; 95% CI = α = 0.01; 99% CI = z7

8 .. \

9 True or False? There is a 95% chance that the true average strength falls between 7.28 and I am 95% confident that the true average strength is captured between 7.28 and At 95% confidence, the values between 7.28 and 8.22 are reasonable values for the true average strength. If I repeated this sampling procedure 100 times, I would expect 95% of the calculated intervals to capture the true average strength. 9

10 Normal Distribution Variance Known Two-sided and one-sided CIs 10

11 Sample size, Confidence Level, and Precision 110CI: Xz Xz( ) % 2 n 2/, / z Width of interval = n2/2/ 2 n Precision = inverse of the width of interval Margin of error = ½ width of interval The larger the sample size, the larger the precision, the smaller the margin of error Trade-off between precision and confidence level A larger confidence level will lead to a less precise interval since will be larger z11

12 Sample Size Determination Suppose you will collect a sample from a population with a Normal distribution and known standard deviation σ. What sample size would you need to obtain a confidence interval for the population mean with a particular margin of error e at significance level α, i.e., confidence level (1- α)100%? 12

13 Example 1 (cont d): CI for μ for different α levels 90% CI: (7.36, 8.14); e = % CI: (7.28, 8.22); e = % CI: (7.13, 8.37); e = Xz1. Xz. 225, 2 / / z196n z. 2 2 e /. / 205z258n z. 2 2 e /. / X 13

14 CI for Population Mean Normal Distribution Variance Unknown Substitute the sample standard deviation for the population standard deviation. 14

15 Student s t distribution N

16 Example 2: Normal Variance Unknown Suppose the following beam strengths (a subset from a previous example) follow a normal distribution N(µ, σ 2 ): 5.9, 7.2, 7.3, 6.3, 8.1, 6.8, 7.0, 7.6, 6.8, 6.5, 7.0, 6.3, 7.9, 9.0, 8.2, 8.7, 7.8, 9.7, 7.4, 7.7, 9.7, 7.8, 7.7, 10.7,

17 Example 2 (Cont d): CI for μ t with d.f. = 24 x 25

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19 Normal Distribution Variance Unknown Two-sided and one-sided CIs Can define upper confidence bound and lower confidence bound similarly as in the variance known case. 19

20 ~NapproximatelyCI for Population Mean in ANY Population with Large Sample Size CLT: Suppose X 1, X n are independent random draws from a population with mean μ and variance σ 2. If n is sufficiently large, then Xn2, n How large is sufficiently large? Rule of thumb: n 30 Is this rule of thumb still applies for constructing intervals. 20

21 Large Sample CI for Population Mean 2X~N aproximat n, n Rule of thumb: n 40, which is larger than for CLT due to additional variability introduced by using S in place of σ. Can construct one-sided CIs as before.ely21

22 Sample Size Calculation n z 2 S / 2 e Can sample size be reasonably determined? Educated guess of S based on a pilot study Educated guess of range (R) of data; if population is not too skewed, estimate S by R/4. 22

23 Example 3: Large Sample CI for μ Suppose he number of daily spam s coming to a mailbox follows a Poisson distribution with intensity parameter λ. The following table gives the observations in 60 days (independent from day to day). # Spam Freq (days) Percent Number of Spam s Bars are the observed; Red dots are the fit with Poisson distribution. 23

24 Example 3: Large Sample CI (Cont d) Construct a 85% two-sided CI 24

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26 Summary of Steps in Constructing CI for a Parameter θ (i.e. mean) 1. Find point estimator (statistic) for parameter 2. Find the (approximate) distribution of the (standardized) statistic 3. Using the sample distribution to find the upper and lower limits (critical values for significant level α) 4. Solve for θ to get the (random) CI 26