BUS304 – Chapter 7 Estimating Population Mean 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? $10$20$15$30$3 $20$0$15$20$7

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 What do you think about these two pieces of information? What does this estimation mean?

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

BUS304 – Chapter 7 Estimating Population Mean 9 When population  is known  Reminder: x1x1 Most of the time, you can get good sample x2x2 x But sometimes, the sample is not good. (unlucky) 95% ? ? z

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

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)(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: t 0 t (df = 5) t (df = 13) Standard Normal (t with df =  )

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 Conf. Level One Tail Two Tail df values of t

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 Determine the 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