Chapter 8 Lesson 8.2 Sampling Variability and Sampling Distributions 8.2: The Sampling Distribution of a Sample Mean

The campus of Pasadena City College has a fish pond. Suppose there are 20 fish in the pond. The lengths of the fish (in inches) are given below: We caught fish with lengths 6.3 inches, 2.2 inches, and 13.3 inches. x = 7.27 inches Let ’ s catch two more samples and look at the sample means. 2 nd sample - 8.5, 4.6, and 5.6 inches. x = 6.23 inches 3 rd sample – 10.3, 8.9, and 13.4 inches. x = inches Suppose we randomly catch a sample of 3 fish from this pond and measure their length. What would the mean length of the sample be? This discrepancy is an example of _______ _______. This is a _______! The true mean  = 8. Notice that some sample means are closer and some farther away; some above and some below the mean.

Fish Pond Continued There are 1140 ( 20 C 3 ) different possible samples of size 3 from this population. If we were to catch all those different samples and calculate the mean length of each sample, we would have a distribution of all possible x. This would be called the …

General Properties of Sampling Distributions of x Rule 1: Rule 2:

The paper “ Mean Platelet Volume in Patients with Metabolic Syndrome and Its Relationship with Coronary Artery Disease ” (Thrombosis Research, 2007) includes data that suggests that the distribution of platelet volume of patients who do not have metabolic syndrome is approximately normal with mean  = 8.25 and standard deviation  = We can use a computer to generate random samples from this population. We will generate 500 random samples of n = 5 and compute the sample mean for each.

Platelets Continued... Similarly, we will generate 500 random samples of n = 10, n = 20, and n = 30. The density histograms below display the resulting 500 x for each of the given sample sizes. What do you notice about the means of these histograms? What do you notice about the standard deviation of these histograms? What do you notice about the shape of these histograms?

General Properties Continued... Rule 3: When the population distribution is normal, the sampling distribution of x is also normal for any sample size n.

The paper “Is the Overtime Period in an NHL Game Long Enough?” (American Statistician, 2008) gave data on the time (in minutes) from the start of the game to the first goal scored for the 281 regular season games from the season that went into overtime. The density histogram for the data is shown below. Let’s consider these 281 values as a population. The distribution is strongly positively skewed with mean m = 13 minutes and with a median of 10 minutes. Using a computer, we will generate 500 samples of the following sample sizes from this distribution: n = 5, n = 10, n = 20, n = 30.

These are the density histograms for the 500 samples Are these histograms centered at approximately  = 13? What do you notice about the standard deviations of these histograms? What do you notice about the shape of these histograms?

General Properties Continued... Rule 4: Central Limit Theorem When n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution is not itself normal. CLT can safely be applied if n exceeds 30. How large is “sufficiently large” anyway?

The Process Going Into the Sampling Distribution Model

A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with  =.16 oz. Sixteen cans are randomly selected, and the soda volume is determined for each one. Let x = the resulting sample mean soda. If the bottler ’ s claim is correct, then the sampling distribution of x is normally distributed with:

.8185 Soda Problem Continued... What is the probability that the mean volume of the sample of 16 sodas is between ounces and ounces? P(11.96 < x < 12.08) =

A hot dog manufacturer asserts that one of its brands of hot dogs has a average fat content of 18 grams per hot dog with standard deviation of 1 gram. Consumers of this brand would probably not be disturbed if the mean was less than 18 grams, but would be unhappy if it exceeded 18 grams. An independent testing organization is asked to analyze a random sample of 36 hot dogs. Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer ’ s claim is incorrect? Since the sample size is greater than 30, the Central Limit Theorem applies. So the distribution of x is approximately normal with

Hot Dogs Continued... Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer ’ s claim is incorrect? P(x > 18.4) =.0082 Values of x at least as large as 18.4 would be observed only about.82% of the time. The sample mean of 18.4 is large enough to cause us to doubt that the manufacturer ’ s claim is correct.

Practice Problem Wildlife scientists studying a certain species of rabbits know that past records indicate they should weigh an average of 118 grams with a standard deviation of 14g. The researchers collect a random sample of 50 bunnies and weigh them. In their sample the mean weight was only 110g. One of the scientists is alarmed, fearing that environmental changes may be adversely affecting the bunnies! Do you think this sample result is unusually low? Explain.

Homework Pg.420: #8.14, 15, 16, 17, 19, 20, 25