Mean (Sample Mean Is an Unbiased Estimator of the Population Mean)

Math 217 Name: ______11-15-05 Quiz for Section 5.2

5.2 Formulas: Distribution of the Sample Mean The sample mean x of an SRS of size n drawn from a large population with mean  and standard deviation  has a sampling distribution with the following properties:

 mean x   (sample mean is an unbiased estimator of the population mean)

 standard deviation  x   / n (to cut the variability of the sample mean in half, quadruple the sample size).  If n is large then x is approximately normally distributed (this is the Central Limit Theorem).  If the original variable X is normally distributed, then x is normally distributed.

1. (8 pts) Children in kindergarten are sometimes given the Ravin Progressive Matrices Test (RPMT) to assess their readiness for learning. Experience at Southwark Elementary School suggests that the RPMT scores for its kindergarten pupils have mean 13.6 and standard deviation 3.1. The distribution is close to normal.

Mr. Lavin has 25 children in his kindergarten class this year. Pretend Mr. Lavin’s class is a SRS of size 25 from the population.

(a) Find the probability that the average score for Mr. Lavin’s class is less than 13.

(b) What is the level L so that there is probability only 0.05 that the average score for Mr. Lavin’s class is less than L? 2. (2 pts) A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of youth of ages 13 to 16 yers. The researchers will report the mean x from their sample as an estimate of the mean cholesterol level μ in this population. The sample result x is an unbiased estimator of the population truth μ no matter what size SRS the study chooses. Explain why a large sample gives more trustworthy results than a small sample.