1. Note 11 of 5E Statistics with Economics and Business Applications Chapter 7 Estimation of Means and Proportions Small-Sample Estimation of a Population Mean

2. Note 11 of 5E Review Review I. What’s in last lecture? Large-Sample Estimation Chapter 7 II. What's in this lecture? Small-Sample Estimation of a Population Mean Read Chapter 7

3. Note 11 of 5EIntroduction When the sample size is small, the estimation procedures in Lecture note 10 are not appropriate. Point estimators remain the same There are small sample interval estimators/confidence intervals for , the mean of a normal population      the difference between two normal population means

4. Note 11 of 5E The Sampling Distribution of the Sample Mean When we take a sample from a normal population, the sample mean has a normal distribution for any sample size n, and has a standard normal distribution. is not normalBut if  is unknown, and we must use s to estimate it, the resulting statistic is not normal.

5. Note 11 of 5E Student’s t Distribution Student’s t distribution, n-1 degrees of freedom.Fortunately, this statistic does have a sampling distribution that is well known to statisticians, called the Student’s t distribution, with n-1 degrees of freedom. We can use this distribution to create estimation procedures for the population mean .

6. Note 11 of 5E Properties of Student’s t degrees of freedom, n-1.Shape depends on the sample size n or the degrees of freedom, n-1. As n increases the shapes of the t and z distributions become almost identical. Mound-shapedMound-shaped and symmetric about 0. More variable than zMore variable than z, with “heavier tails”

7. Note 11 of 5E t distribution

8. Note 11 of 5E Using the t-Table Table 4 gives the values of t that cut off certain critical values in the right tail of the t distribution. df a t a,Use index df and the appropriate tail area a to find t a, the value of t with area a to its right. For a random sample of size n = 10, find a value of t that cuts off.025 in the right tail. Row = df = n –1 = 9 t.025 = 2.262 Column subscript = a =.025 a tatatata

9. Note 11 of 5E Small Sample Confidence Interval for Population Mean  Assumption: population must be normal

10. Note 11 of 5E Example Ten randomly selected students were each asked to list how many hours of television they watched per month. The results are 8266908475 88809411091 Find a 90% confidence interval for the true mean number of hours of television watched per month by students.

11. Note 11 of 5E Example Continued We find the critical t value of 1.833 by looking on the t table in the row corresponding todf= 9, in the column with label t. 050. The 90% confidence interval for is Calculating the sample mean and standard deviation we have 842.11sand,86x,10n  Calculating the sample mean and standard deviation we have 842.11sand,86x,10n  n s tx  842.11 )833.1(86 .6  ).92,14.79(

12. Note 11 of 5E Estimating the Difference between Two Means You can also create a 100(1-  )% confidence interval for  1 -  2. Remember the three assumptions: 1.Original populations normal 2.Samples random and independent 3.Equal population variances Remember the three assumptions: 1.Original populations normal 2.Samples random and independent 3.Equal population variances

13. Note 11 of 5E Example A student recorded the mileage he obtained while commuting to school in his car. He kept track of the mileage for twelve different tanks of fuel, involving gasoline of two different octane ratings. Compute the 95% confidence interval for the difference of mean mileages. His data follow: 87 Octane90 Octane 26.4, 27.6, 29.730.5, 30.9, 29.2 28.9, 29.3, 28.831.7, 32.8, 29.3

14. Note 11 of 5E Example Let 87 octane fuel be the first group and 90 octane fuel the second group, so we have n 1 = n 2 = 6 and d.f.=n 1 +n 2 -2=10. The critical value of t is 2.228.

15. Note 11 of 5E Key Concepts I. One Population Mean

16. Note 11 of 5E Key Concepts II.Two Population Means 1. Both populations are normal. 2. Two samples are independent. 3. Two populations have a common variance.