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Section 6-4Sampling Distributions

and Estimators

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The height of young women varies approximately according to the N(64.5, 2.5) distribution. The random variable measured (X) is the height of a randomly selected young woman. In this Activity you will use the TI83 to sample from this distribution and them use Post-it notes to construct a distribution of averages.

1. If we choose one woman at random, the heights we get in repeated choices follow the N(64.5, 2.5) distribution. On your calculator, go into the Stats/List editor and clear L1. Simulate the heights of 100 randomly selected young women and store these heights in L1:* Place your cursor at the top of L1; Press Math, PRB, Choose 6: randNorm( and complete the command: randNorm (64.5, 2.5, 100) and press ENTER.

2. Plot a histogram of the 100 heights by deselecting active fxn in the Y= window, and turn off all STAT PLOTS. Set WINDOW dimensions to X[57,72] (Xscl:2.5) and Y[-10,45] (Yscl:5) to extend 3 standard devs of the mean. Define Plot 1 to be a histogram using the heights in L1. Press GRAPH to plot the histogram. Is it fairly symmetric or clearly skewed?

3. Use 1-var stats to find the mean, median, and standard deviation for your data. Compare x-bar with the population mean of 64.5. Compare the sample standard deviation with the population std. dev. Of 2.5. How do the mean and median for your 100 heights compare? Recall that the close the mean and the median are, the more symmetric the distribution.

4. Define Plot2 to be a boxplot using L1, and press GRAPH. The boxplot will be plotted above the histogram. Does the boxplot appear symmetric? How close is the median in the boxplot to the mean of the histogram? Based on the appearance of the histogram and the boxplot, and a comparison of the mean and median, would you say that the distribution is nonsymmetric, moderately symmetric, or very symmetric?

5. Write the mean for your sample on a Post-it Note. Put your post it note on the appropriate location on the graph on the board. When the “Post-it” histogram is complete:* What is the approximate shape of the distribution of x-bar:* What is the center and std. dev. of x-bar? How does this compare with the mean and std. dev. for the heights of all young women?

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Key Concept

The main objective of this section is to understand the concept of a sampling distribution of a statistic, which is the distribution of all values of that statistic when all possible samples of the same size are taken from the same population.

We will also see that some statistics are better than others for estimating population parameters.

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Definitions The sampling distribution of a statistic (such as the sample proportion or sample mean) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.

The value of a statistic, such as the sample mean x, depends on the particular values included in the sample, and generally varies from sample to sample. This variability of a statistic is called sampling variability.

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The sampling distribution of a proportion is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.

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The sampling distribution of the mean is the distribution of sample means, with all samples having the same sample size n taken from the same population. (The sampling distribution of the mean is typically represented as a probability distribution in the format of a table, probability histogram, or formula.)

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Ex. 1

Take a sample of 20 statistics students at the high school level and calculate the percentage of students who got a B or Higher on the chapter 1 test. Take another 20 statistics students, and calculate the percentage of students who got a B or Higher on the chapter 1 test. Keep going until we have taken all possible samples from the high school population in the USA (this percentage will vary from sample to sample).

Draw a histogram of the distribution of proportions…. The mean of these proportions will equal the USA

percentage of students who got a B or higher on the Chapter 1 test.

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Ex. 2

Take a sample of 20 statistics students at the high school level and calculate the mean height of students. Take another 20 statistics students, and calculate the mean height of students. Keep going until we have taken all possible samples from the high school population in the USA (this mean will vary from sample to sample).

Draw a histogram of the distribution of means….The mean of this distribution will equal the USA

mean of student heights.

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Example When two births are randomly selected, the sample space is:

{bb, bg, gb, gg} Those four equally likely outcomes suggest the following probability

distribution for the number of girls from 2 births:

Here is the sampling distribution for the proportion of girls:

We usually describe a sampling distribution using a table that lists values of the sample statistic along with their corresponding probabilities.

X 0 1 2

P(x) 0.25 0.50 0.25

Proportion of girls from 2 births

0/2 = 0 ½ = 0.5 2/2 = 1

P(x) 0.25 0.50 0.25

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Example Consider the genders of the Senators in the 107th

Congress. There are only 100 members, 13 Females and 87 Males.

The population proportion of females is:

Usually, we don’t know all of the members of the population, so we must estimate it from a sample.

Sample 1: M F M M F M M M M M

Sample 2: M F M M M M M M M M

Sample 3: M M M M M M F M M M

Sample 4: M M M M M M M M M M

Sample 5: M M M M M M M M F M

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Example (cntd…)

Suppose we take another 95 samples (for a total of 100). Combining these additional samples with the first 5, we get 100 samples summarized here

If we were to include all other possible sample sizes of 10 (all 100,000,000,000,000,000,000 of them!) the mean of the sample proportions would equal 0.13!

Proportion of Female Senators

Frequency P(x) Sample

Proportion

0.0 26 26/100

0.1 41 41/100

0.2 24 24/100

0.3 7 7/100

0.4 1 1/100

0.5 1 1/100

Mean 0.119

Std.dev. 0.100

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Properties

Sample proportions tend to target the value of the population proportion. (That is, the

mean of all possible sample proportions = mean of the population proportion.)

Under certain conditions, the distribution of the sample proportion can be approximated by a normal distribution.

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Estimators

Some statistics work much better than others as estimators of the population. The example that follows shows this.

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Example - Sampling Distributions

A population consists of the values 1, 2, and 5. We randomly select samples of size 2 with replacement. There are 9 possible samples.

What are they?

a. For each sample, find the mean, median, range, variance, and standard deviation.

b. For each statistic, find the mean from part (a)

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Fill in the table (part a) (For each sample, find

the mean, median, range, variance, and standard deviation).

Sample Mean Median Range Variance Std. Dev.

Prop. Of odd #s

Prob.

1,1

1,2

1,5

2,1

2,2

2,5

5,1

5,2

5,5

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Fill in the bottom row (part b): (For each statistic, find the mean from part (a))

Mean of statistics : ___ ___ ___ ___ ____ ____ Population Parameter: ___ ___ ___ ___ ____ ____ Does the sample statistic target the population parameter?

____ ___ ___ ___ ___ ____

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Interpretation of Sampling Distributions

We can see that when using a sample statistic to estimate a population parameter, some statistics are good in the sense that they target the population parameter and are therefore likely to yield good results. Such statistics are called unbiased estimators.

Statistics that target population parameters: mean, variance, proportion

Statistics that do not target population parameters: median, range, standard deviation

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Recap

In this section we have discussed:

Sampling distribution of a statistic.

Sampling distribution of a proportion.

Sampling distribution of the mean.

Sampling variability.

Estimators.