1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Loucks St. Edward’s University SLIDES. BY

2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The sampling distribution of is the probability The sampling distribution of is the probability distribution of all possible values of the sample mean. Sampling Distribution of where:  = the population mean E ( ) =  Expected Value of Expected Value of When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased.

3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of  = the standard deviation of  = the standard deviation of the population n = the sample size Standard Deviation of Standard Deviation of

4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Central Limit Theorem When the population from which we are selecting When the population from which we are selecting a random sample does not have a normal distribution, the central limit theorem is helpful in identifying the shape of the sampling distribution of. CENTRAL LIMIT THEOREM CENTRAL LIMIT THEOREM In selecting random samples of size n from a In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal mean can be approximated by a normal distribution as the sample size becomes large. distribution as the sample size becomes large. CENTRAL LIMIT THEOREM CENTRAL LIMIT THEOREM In selecting random samples of size n from a In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal mean can be approximated by a normal distribution as the sample size becomes large. distribution as the sample size becomes large.

5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of E ( ) = 

6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. Point Estimate +/  Margin of Error The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. Margin of Error and the Interval Estimate

7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is Margin of Error and the Interval Estimate

8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimate of a Population Mean:  Unknown If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . This is the  unknown case. This is the  unknown case. In this case, the interval estimate for  is based on the t distribution. In this case, the interval estimate for  is based on the t distribution. n (We’ll assume for now that the population is normally distributed.)

9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. William Gosset, writing under the name “Student”, William Gosset, writing under the name “Student”, is the founder of the t distribution. is the founder of the t distribution. William Gosset, writing under the name “Student”, William Gosset, writing under the name “Student”, is the founder of the t distribution. is the founder of the t distribution. t Distribution Gosset was an Oxford graduate in mathematics and Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin. worked for the Guinness Brewery in Dublin. Gosset was an Oxford graduate in mathematics and Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin. worked for the Guinness Brewery in Dublin. He developed the t distribution while working on He developed the t distribution while working on small-scale materials and temperature experiments. small-scale materials and temperature experiments. He developed the t distribution while working on He developed the t distribution while working on small-scale materials and temperature experiments. small-scale materials and temperature experiments.

10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. t Distribution A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. computation of s. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. computation of s.

11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. t Distribution A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion. less dispersion. A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion. less dispersion. As the degrees of freedom increases, the difference As the degrees of freedom increases, the difference between the t distribution and the standard normal between the t distribution and the standard normal probability distribution becomes smaller and probability distribution becomes smaller and smaller. smaller. As the degrees of freedom increases, the difference As the degrees of freedom increases, the difference between the t distribution and the standard normal between the t distribution and the standard normal probability distribution becomes smaller and probability distribution becomes smaller and smaller. smaller.

12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. t Distribution Standardnormaldistribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) of freedom) 0 z, t

13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. t Distribution The standard normal z values can be found in the The standard normal z values can be found in the infinite degrees (  ) row of the t distribution table. infinite degrees (  ) row of the t distribution table. The standard normal z values can be found in the The standard normal z values can be found in the infinite degrees (  ) row of the t distribution table. infinite degrees (  ) row of the t distribution table.

14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. t Distribution Standard normal z values

15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Estimate where: 1 -  = the confidence coefficient t  /2 = the t value providing an area of  /2 t  /2 = the t value providing an area of  /2 in the upper tail of a t distribution in the upper tail of a t distribution with n - 1 degrees of freedom with n - 1 degrees of freedom s = the sample standard deviation s = the sample standard deviation Interval Estimate of a Population Mean:  Unknown

16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A reporter for a student newspaper is writing an A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 efficiency apartments within a half-mile of campus resulted in a sample mean of $750 per month and a sample standard deviation of $55. Interval Estimate of a Population Mean:  Unknown n Example: Apartment Rents Let us provide a 95% confidence interval estimate Let us provide a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half-mile of campus. We will assume this population to be normally distributed.

17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. At 95% confidence,  =.05, and  /2 =.025. In the t distribution table we see that t.025 = t.025 is based on n  1 = 16  1 = 15 degrees of freedom. Interval Estimate of a Population Mean:  Unknown

18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. We are 95% confident that the mean rent per month We are 95% confident that the mean rent per month for the population of efficiency apartments within a half-mile of campus is between $ and $ n Interval Estimate Interval Estimate of a Population Mean:  Unknown Margin of Error

19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimate of a Population Mean:  Unknown n Adequate Sample Size If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended. If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended. In most applications, a sample size of n = 30 is In most applications, a sample size of n = 30 is adequate when using the expression to adequate when using the expression to develop an interval estimate of a population mean. develop an interval estimate of a population mean. In most applications, a sample size of n = 30 is In most applications, a sample size of n = 30 is adequate when using the expression to adequate when using the expression to develop an interval estimate of a population mean. develop an interval estimate of a population mean.

20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimate of a Population Mean:  Unknown n Adequate Sample Size (continued) If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice.

21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is Interval Estimate of a Population Proportion

22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimate of a Population Proportion The sampling distribution of plays a key role in The sampling distribution of plays a key role in computing the margin of error for this interval computing the margin of error for this interval estimate. estimate. The sampling distribution of plays a key role in The sampling distribution of plays a key role in computing the margin of error for this interval computing the margin of error for this interval estimate. estimate. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5.

23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.  /2 Interval Estimate of a Population Proportion n Normal Approximation of Sampling Distribution of Sampling distribution of Sampling distribution of p p 1 -  of all values 1 -  of all values

24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Estimate Interval Estimate of a Population Proportion where: 1 -  is the confidence coefficient z  /2 is the z value providing an area of z  /2 is the z value providing an area of  /2 in the upper tail of the standard  /2 in the upper tail of the standard normal probability distribution normal probability distribution is the sample proportion is the sample proportion

25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Political Science, Inc. (PSI) specializes in voter polls Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, PSI interviewers ask Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day. Interval Estimate of a Population Proportion n Example: Political Science, Inc.

26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. In a current election campaign, PSI has just found In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor the candidate. Interval Estimate of a Population Proportion n Example: Political Science, Inc.

27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. where: n = 500, = 220/500 =.44, z  /2 = 1.96 Interval Estimate of a Population Proportion PSI is 95% confident that the proportion of all voters that favor the candidate is between.3965 and PSI is 95% confident that the proportion of all voters that favor the candidate is between.3965 and =