sampling_dist_avg_key

PHW142 Practice Problems Using The Normal Curve For Sums And Averages KEY 1: Blood Cholesterol based on Baldi and Moore exercise 13.5 (both editions) A government sample survey plans to measure the blood cholesterol levels of a simple random sample (SRS) of men aged 20 to 34 years. The researchers will report the mean ¯ x from their sample as an estimate of the mean cholesterol level µ in this population. 1a Explain to someone who knows no statistics what it means to say that ¯ x is an “unbiased” of µ 1a answer In the long-run, the average of the values of the averages ¯ x from many samples is equal to µ . As Baldi and Moore say, ’ ¯ x is not systematically higher or lower than µ .’ 1b The sample result ¯ x is an unbiased estimator of the true population µ no matter what size SRS the study uses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample. [Think about the concentration of the sample averages ¯ x around the mean µ .] 1b answer The sampling distribution of ¯ x is much more concentrated around µ for large samples. We’ll use the term ’precise’ to describe such estimates. As Baldi and Moore say, more informally, ’With large samples, ¯ x is more likely to be close to µ .’ 1

2: More On Blood Cholesterol based on Baldi and Moore exercise 13.7 (both editions) Suppose that in fact the blood cholesterol levels of all men aged 20 to 34 years follows the normal distribution with mean µ = 188 milligrams per deciliter (mg/dl) and standard deviation σ = 41 mg/dl. 2a What is the probability that an individual selected at random from this population has a blood cholesterol level between 185 and 191 mg/dl ? 2a answer pnorm ( 191 , mean = 188 , sd = 41 , lower.tail= TRUE ) - pnorm ( 185 , mean = 188 , sd = 41 , lower.tail= TRUE ) ## [1] 0.05832974 2b Choose a simple random sample (SRS) of 100 men from this population. What are the mean and standard deviation of the sampling distribution of ¯ x ? What is its shape, and how do you know the shape? 2b answer mean of the sampling distribution of ¯ x is the mean of the population, 188 mg/dl standard deviation of ¯ x is 41 √ 100 = 4.1 mg/dl shape = Normal curve, as the distribution of the blood cholesterol levels is normal curve. 2c For our SRS of 100 men, what is the probability that ¯ x takes a value between 185 and 191 mg/dl? (Another way to describe this probability is that ¯ x estimates µ within ± 3 mg/dl.) 2c answer pnorm ( 191 , mean = 188 , sd = 4.1 , lower.tail= TRUE ) - pnorm ( 185 , mean = 188 , sd = 4.1 , lower.tail= TRUE ) ## [1] 0.5356528 the probability that an SRS of size 100 from this population has an average blood cholesterol level between 185 and 191 mg/dl is about .5358 2