Section 6 4 Sampling Distributions and Estimators Definition

Section 6. 4 Sampling Distributions and Estimators

Definition The sampling distribution of the mean is the distribution of sample means, with all samples having the sample size n taken from the same population. (The sampling distribution of the mean is typically represented as a dot plot or histogram. ) In other words: Take all samples of size n from a group of N people and find the mean height of each sample. Make a dot plot to display these means. You will then find the mean of all the dots on the dot plot. This number will equal or be very close to the actual mean of the N people. The dot plot is the graphic display of the sampling distribution and the mean of the dots is the mean of the sampling distribution.

Example 1 - Sampling Distributions Specific results from 10, 000 trials All outcomes are equally likely so the population mean is 3. 5; the mean of the 10, 000 trials is 3. 49. If continued indefinitely, the sample mean will be 3. 5. Also, notice the distribution is “normal. ”

Notation We need to distinguish between a population mean µ and some sample mean: µ = population mean = sample mean

Activity 1: Heights 4 volunteers, please Initials Height (inches) Person 1: Person 2: Person 3: Person 4: Initials Mean height / / / / What is the mean of these means? / /

Definition The sampling distribution of the proportion is the distribution of sample proportions, with all samples having the sample size n taken from the same population. In other words, Take all samples of size 20 from a group of 200 Reese’s Pieces and find the proportion of orange candies in each sample and making a dot plot to display these proportions. You will then find the mean proportion value of all the dots on the dot plot. This number will equal or be very close to the actual proportion of orange candies. The dot plot is the graphic display of the sampling distribution and the mean of the sample proportions of the dots is the mean of the sampling distribution. Reeses. Pieces

Example 2 - Sampling Distributions Specific results from 10, 000 trials All outcomes are equally likely so the population proportion of odd numbers is 0. 50; the proportion of the 10, 000 trials is 0. 50. If continued indefinitely, the mean of sample proportions will be 0. 50. Also, notice the distribution is “approximately normal. ”

Notation We need to distinguish between a population proportion p and some sample proportion: p = population proportion = sample proportion (pronounced “p hat”)

Activity 2: Candy What is the proportion of orange Reese’s to the amount of total Reese’s you have? What is the mean of these proportions?

Unbiased Estimators Sample means, variances, and proportions are unbiased estimators, that is they target the population parameter. (They have a mean that is equal to the mean of the corresponding population. ) These statistics are better in estimating the population parameter.

Biased Estimators Sample medians, ranges and standard deviations are biased estimators, that is they do NOT target the population parameter. Note: the bias with the standard deviation is relatively small in large samples so s is often used to estimate.

Example 3 Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10. Consider the values 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from a population that is normally distributed. Show that medians are biased estimators. a) b) c) d) List all possible samples and their medians. Describe the sampling distribution of the medians in the format of a table summarizing the probability distribution Based on the results, do the sample medians target the population median? What do these results indicate about the sample median as an estimator of the population median?

a) List all possible samples and their medians. Outcomes Medians 2, 2 2, 3 2 2. 5 2, 10 6 3, 2 2. 5 3, 3 3 3, 10 6. 5 10, 2 6 10, 3 6. 5 10, 10 10

b) Describe the sampling distribution of the medians in the format of a table summarizing the probability distribution Outcomes Medians 2, 2 2, 3 2, 10 3, 2 3, 3 3, 10 10, 2 10, 3 10, 10 2 2. 5 6 2. 5 3 6. 5 6 6. 5 10 Sample Median 2 2. 5 3 6 6. 5 10 Probability

c) Based on the results, do the sample medians target the population median? Population Median: Vs. Mean of Sample Medians: d) What do these results indicate about the sample median as an estimator of the population median?