1 / 20

Review – Last Week

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?.

ros
Télécharger la présentation

Review – Last Week

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related