BUSN225 - Week 8.2 - Chapter 7 - sample methods and the central limit theorem

BUSN225 Research Methods and Data Analysis 1 Week 8 Chapter 7: Sampling Methods and the Central Limit Theorem Sampling Methods What are the 4 key methods of sampling? 1) ___________________________________ 2) ___________________________________ 3) ___________________________________ 4) ___________________________________ Sampling Error Sampling Error is the difference between a _________________ and its corresponding ________________________. Central limit theorem If random samples of n trials, where n > 30 , are drawn from a non-normally distributed population with a finite mean μ and standard deviation σ , then the distribution of the sample mean, ´ x , is approximately normally distributed, with mean μ and standard deviation σ √ n Practical Rules for Real Applications and the Use of CLT 1) Original population is normally distributed 2) Sample size is n > 30

If either one of these rules applies, then the distribution of the sample means ´ x , has the following parameters: Sampling Distribution of the Sample Mean Mean of sample means Standard Deviation of sample means Z-value to compute probabilities μ ´ x = μ σ ´ x = σ √ n z = ´ x − μ σ √ n Sampling Distribution of the Sample Proportion When np and n ( 1 − p )> 5 , the sampling distribution of the sampling proportion will be approximately normally distributed with the following parameters: Sample Proportion Mean of the sampling proportion Standard Error of the sampling proportion Z-value to compute probabilities ´ p = x n μ = p σ p = √ p ( 1 − p ) n . z = ´ p − p √ p ( 1 − p ) n

2) The duration of Alzheimer’s disease from the onset of symptoms until death ranges from 3 to 20 years. It follows a normal distribution. The average is 8 years and the standard deviation is 4 years. Medical records of 30 deceased Alzheimer’s patients were sampled. a) Determine the probability that the average duration is less than 7 years. b) Determine the probability that the average duration lies within one year of the population mean. 3) Let x represent the scores of a mathematics achievement test given to Grade 8 students. Suppose that for the population of all test scores, the mean is 50 and the

standard deviation of 15. If a random sample of 100 values of x is to be obtained, find the probability that the sample mean ´ x will be between 51 and 53. 4) The mean wage of the employees of a company is $42,500 and the standard deviation is $2000. What is the probability that the mean wage of 75 randomly selected workers will exceed $43,000? 5) An important expectation of a federal income tax reduction is that consumers will reap a substantial portion of the tax savings. Suppose estimates of the portion of total tax saved, based on a random sampling of 35 economists, have a mean of 26% and a

The foreman samples 60 bags with each shipment and if the bags average more than 6 percent underweight, the whole shipment is returned to the supplier. a) What is the probability that in a sample of 160 bags, the foreman will find that less than 4 percent are underweight? b) What is the probability that in a sample of 200, the foreman will find more than 5% of the bags underweight?

8) A daycare provider has determined that 80% of the children in her care develop colds each winter. This year, there are 40 children in the daycare. What is the probability that fewer than 30 children will develop colds this winter?