Chapter 12 Inference About One Population

We shall develop techniques to estimate and test three population parameters.  Population mean   Population variance  2  Population proportion p Introduction

Recall that when  is known we use the following statistic to estimate and test a population mean When  is unknown, we use its point estimator s, and the z-statistic is replaced then by the t-statistic Inference About a Population Mean When the Population Standard Deviation Is Unknown

The t - Statistic s 0 The t distribution is mound-shaped, and symmetrical around zero. The “degrees of freedom”, (a function of the sample size) determine how spread the distribution is (compared to the normal distribution) d.f. = v 2 d.f. = v 1 v 1 < v 2 t

How to calculus sample variance

Example 1  In order to determine the number of workers required to meet demand, the productivity of newly hired trainees is studied.  It is believed that trainees can process and distribute more than 450 packages per hour within one week of hiring.  Can we conclude that this belief is correct, based on productivity observation of 50 trainees Testing  when  is unknown

Example 1 – Solution  The problem objective is to describe the population of the number of packages processed in one hour.  The data are interval. H 0 :  = 450 H 1 :  > 450  The t statistic d.f. = n - 1 = 49 We want to prove that the trainees reach 90% productivity of experienced workers We want to prove that the trainees reach 90% productivity of experienced workers Testing  when  is unknown

Solution continued (solving by hand)  The rejection region is t > t ,n – 1 t ,n - 1 = t.05,49  t.05,50 = Testing  when  is unknown

The test statistic is Since 1.89 > we reject the null hypothesis in favor of the alternative. There is sufficient evidence to infer that the mean productivity of trainees one week after being hired is greater than 450 packages at.05 significance level Rejection region Testing  when  is unknown

Estimating  when  is unknown Confidence interval estimator of  when  is unknown

Example 2  An investor is trying to estimate the return on investment in companies that won quality awards last year.  A random sample of 83 such companies is selected, and the return on investment is calculated had he invested in them.  Construct a 95% confidence interval for the mean return. Estimating  when  is unknown

Solution (solving by hand)  The problem objective is to describe the population of annual returns from buying shares of quality award-winners.  The data are interval.  Solving by hand From the data we determine t.025,82  t.025,80 Estimating  when  is unknown

Checking the required conditions We need to check that the population is normally distributed, or at least not extremely nonnormal. There are statistical methods to test for normality (one to be introduced later in the book). From the sample histograms we see …

A Histogram for Example 1 Packages A Histogram for Example 2 Returns

Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean n > 30 ?  known ?  known ? Popul. approx.normal ?  known ? Use s to estimate  Use s to estimate  Increase n to > 30 Yes Yes Yes Yes No No No No

Example 1

Solution

Example 2

Solution

Example 3

Solution

Inference About a Population Variance Sometimes we are interested in making inference about the variability of processes. Examples:  The consistency of a production process for quality control purposes.  Investors use variance as a measure of risk. To draw inference about variability, the parameter of interest is  2.

The sample variance s 2 is an unbiased, consistent and efficient point estimator for  2. The statistic has a distribution called Chi-squared, if the population is normally distributed. d.f. = 5 d.f. = 10 Inference About a Population Variance

Example 3 (operation management application)  A container-filling machine is believed to fill 1 liter containers so consistently, that the variance of the filling will be less than 1 cc (.001 liter).  To test this belief a random sample of 25 1-liter fills was taken, and the results recorded  Do these data support the belief that the variance is less than 1cc at 5% significance level? Testing the Population Variance

Solution  The problem objective is to describe the population of 1-liter fills from a filling machine.  The data are interval, and we are interested in the variability of the fills.  The complete test is: H 0 :  2 = 1 H 1 :  2 <1 We want to know whether the process is consistent Testing the Population Variance

There is insufficient evidence to reject the hypothesis that the variance is less than 1. There is insufficient evidence to reject the hypothesis that the variance is less than 1. Solving by hand –Note that (n - 1)s 2 =  (x i - x) 2 =  x i 2 – (  x i ) 2 /n –From the sample, we can calculate  x i = 24,996.4, and  x i 2 = 24,992,821.3 –Then (n - 1)s 2 = 24,992,821.3-(24,996.4) 2 /25 =20.78 Testing the Population Variance

Rejection region  =  =.95 Do not reject the null hypothesis Testing the Population Variance

Testing and Estimating a Population Variance From the following probability statement P(  2 1-  /2 <  2 <  2  /2 ) = 1-  we have (by substituting  2 = [(n - 1)s 2 ]/  2.)

Example 4

Solution

Example 5 During annual checkups physician routinely send their patients to medical laboratories to have various tests performed. One such test determines the cholesterol level in patients’ blood. However, not all tests are conducted in the same way. To acquire more information, a man was sent to 10 laboratories and in each had his cholesterol level measured. The results are listed here. Estimate with 95% confidence the variance of these measurements

Solution

Inference About a Population Proportion When the population consists of nominal data, the only inference we can make is about the proportion of occurrence of a certain value. The parameter p was used before to calculate these probabilities under the binomial distribution.

Statistic and sampling distribution  the statistic used when making inference about p is: – Under certain conditions, [np > 5 and n(1-p) > 5], is approximately normally distributed, with  = p and  2 = p(1 - p)/n. Inference About a Population Proportion

Testing and Estimating the Proportion Test statistic for p Interval estimator for p (1-  confidence level)

Example 6

Solution

Selecting the Sample Size to Estimate the Proportion Recall: The confidence interval for the proportion is Thus, to estimate the proportion to within W, we can write

Selecting the Sample Size to Estimate the Proportion The required sample size is

12.40 Selecting the Sample Size Two methods – in each case we choose a value for then solve the equation for n. Method 1 : no knowledge of even a rough value of. This is a ‘worst case scenario’ so we substitute =.50 Method 2 : we have some idea about the value of. This is a better scenario and we substitute in our estimated value.