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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: $10 $20 $15 $30 $3 $20 $0 $15 $20 $7 What is the sampling error? What do you think based on this sample? BUS304 – Chapter 7 Estimating Population Mean 1 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 2 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 3 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 4 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 5 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 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% 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 $50,000 in Sep 2005 -- with a confident level of 80% BUS304 – Chapter 7 Estimating Population Mean 6 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 7 Estimation process Random Sample Population (mean, μ, is unknown) Mean x = 50 Sample I am 95% confident that μ is between 40 & 60. BUS304 – Chapter 7 Estimating Population Mean 8 When population is known Reminder: x is normally distributed, with mean μ, and standard deviaiton n 95% ? ? n x μx μ Most of the time, you can get good sample z x1 x2 But sometimes, the sample is not good. (unlucky) BUS304 – Chapter 7 Estimating Population Mean 9 When population is known Formula: the confidence interval x Z 2 n where -- (1 ) *100% is the confidence level; -- Z is the z-score which makes P(0<z<1)= 2 1- ; 2 -- x is the sample mean; -- n is the standard deviation of sample mean. BUS304 – Chapter 7 Estimating Population Mean 10 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 11 Example Problem 7.3 (a) Problem 7.4 (a) (page 281) BUS304 – Chapter 7 Estimating Population Mean 12 When is unknown We use the sample standard deviation to estimate. How to calculate sample standard deviation s ? (check chapter 3) Revised formula x t 2 s n -- s is the sample standard deviation -- t/2 : the cutoff t-value from t-distribution BUS304 – Chapter 7 Estimating Population Mean 13 Student’s t-distribution Standard Normal (t with df = ) t (df = 13) t (df = 5) t 0 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: t x-μ s n BUS304 – Chapter 7 Estimating Population Mean 14 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 Conf. Level 0.7 0.8 0.9 0.95 0.98 0.99 One Tail 0.15 0.1 0.05 0.025 0.01 0.005 Two Tail 0.3 0.2 0.1 0.05 0.02 0.01 BUS304 – Chapter 7 Estimating Population Mean 15 df values of t 8 9 10 1.8125 11 12 Examples Problem 7.3 (b) Problem 7.4 (b) Problem 7.9 (P282) BUS304 – Chapter 7 Estimating Population Mean 16 Determine the sample size When is known: Confidence interval: x Z 2 Z 2 n n 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. Z 2 Z 2 required error n required error n BUS304 – Chapter 7 Estimating Population Mean 17 2 Exercise Problem 7.25 (P. 288) BUS304 – Chapter 7 Estimating Population Mean 18 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 Z 2 n required error 2 to determine the sample size. Since we already have 20, n-20 more is still needed. BUS304 – Chapter 7 Estimating Population Mean 19 Exercise Problem 7.27 BUS304 – Chapter 7 Estimating Population Mean 20