Estimating Population Mean: Sampling and Confidence Intervals

1. Review – Last Week • Sampling error The radio station claims that on average a household in San Diego spends $18 on candy this Halloween. A sample of 10 households reported that their expenditure on candy is as follows: What is the sampling error? What do you think based on this sample? BUS304 – Chapter 7 Estimating Population Mean

2. More review • Assuming the radio station also reports that the variance of the expenditure on candy is 90. • Assuming the report is true, what is the probability that your sample mean is 14 or lower? • What is the potential problem of the probability evaluation method you’ve used? BUS304 – Chapter 7 Estimating Population Mean

3. Chapter 7 Estimating Population Mean • In this chapter, we study how to use sampling result to estimate population mean. • Determine a confidence interval • When population variance is known • When population variance is unknown • Determine the sample size to control the estimation error BUS304 – Chapter 7 Estimating Population Mean

4. Point and Interval Estimates • Point estimate of population mean • Use the sample mean – a single value. Based on a result of clustered sampling, the average housing price in San Diego county is $495,000 in Sep 2005 • Confident Interval • Use a range to estimate. Based on a result of clustered sampling, the average housing price in San Diego county is $495,000  $5,000 in Sep 2005. What does this estimation mean? What do you think about these two pieces of information? BUS304 – Chapter 7 Estimating Population Mean

5. Confidence Intervals • Usually has the sample mean as the middle point • Is usually associated with a confidence level. • What is the probability that the population mean is in the range • In other words, how confident are you? • It provides more information about a population characteristic than does a point estimate BUS304 – Chapter 7 Estimating Population Mean

6. Determine confidence intervals • The estimation is normally a tradeoff between confidence interval and confidence level. • The larger the interval, the higher the confidence level. – but less useful • The smaller the interval, the lower the confidence level. – less accurate Based on a result of clustered sampling, the average housing price in San Diego county is $495,000  $400,000 in Sep 2005 -- with a confident level of 99% Based on a result of clustered sampling, the average housing price in San Diego county is $495,000  $50,000 in Sep 2005 -- with a confident level of 80% BUS304 – Chapter 7 Estimating Population Mean

7. Determine the confidence level • In real life practice, the required confidence level is normally given: E.g. give your estimation about the average annual income per household in San Diego county with 90% confidence level The confidence level is always lower than 100% Never 100% sure BUS304 – Chapter 7 Estimating Population Mean

8. Population Mean x = 50 (mean, μ, is unknown) Sample I am 95% confident that μ is between 40 & 60. Estimation process Random Sample BUS304 – Chapter 7 Estimating Population Mean

9. 95% ? ? x z x1 x2 When population  is known • Reminder: Most of the time, you can get good sample But sometimes, the sample is not good. (unlucky) BUS304 – Chapter 7 Estimating Population Mean

10. When population  is known • Formula: the confidence interval where BUS304 – Chapter 7 Estimating Population Mean

11. Steps to determine the confidence interval • Step 1: check whether sample mean x is given • If not, compute it. • Step 2: check whether the standard deviation of sample mean is given • Sometimes, only population standard deviation  is given. Divided it by n then. • Step 3: use the required confidence level to compute z/2 • Confidence level = 1- • Probability = (1-)/2 • Check the reverse table for z/2 BUS304 – Chapter 7 Estimating Population Mean

12. Example • Problem 7.3 (a) • Problem 7.4 (a) (page 281) BUS304 – Chapter 7 Estimating Population Mean

13. When  is unknown • We use the sample standard deviation to estimate. • How to calculate sample standard deviation s? (check chapter 3) • Revised formula -- s is the sample standard deviation -- t/2 : the cutoff t-value from t-distribution BUS304 – Chapter 7 Estimating Population Mean

14. Student’s t-distribution • A set of bell-shaped symmetric distributions • Each has a degree of freedom: d.f. • When df increases, the t-distribution gets closer to normal distribution • Formula for degree of freedom: d.f. = n-1 • t-value for each x: Standard Normal (t with df = ) t (df = 13) t (df = 5) t 0 BUS304 – Chapter 7 Estimating Population Mean

15. Get the cut-off t-value from the table • Use the table on page 597 “Values of t for selected probabilities” • To check the table: • First, get the degree of freedom. e.g. d.f. =10 • And the confident level (e.g. 90%) • When d.f. gets too large, use normal table df values of t 1.8125 BUS304 – Chapter 7 Estimating Population Mean

16. Examples • Problem 7.3 (b) • Problem 7.4 (b) • Problem 7.9 (P282) BUS304 – Chapter 7 Estimating Population Mean

17. Determinethe sample size • When  is known: • Confidence interval: • is called the estimation error • Sometimes, the estimation error is required not to be too large • Also, the confidence level (1-) is also required • You have to get the large enough sample to guarantee you meet both requirement. BUS304 – Chapter 7 Estimating Population Mean

18. Exercise • Problem 7.25 (P. 288) BUS304 – Chapter 7 Estimating Population Mean

19. When  is unknown • Need more complicated procedure • Pilot sample: (Page 287) • Start using a sample of n= 10 or 20. • Get the sample mean and sample standard deviation • Use the sample standard deviation to estimate the population standard deviation. s • Use to determine the sample size. • Since we already have 20, n-20 more is still needed. BUS304 – Chapter 7 Estimating Population Mean

20. Exercise • Problem 7.27 BUS304 – Chapter 7 Estimating Population Mean