Warm-up7.2 Generating Sampling Distributions

Answers to Warm-up

(C) X is close to the mean and so will have a z-score closeto 0. Boxplots, if they show an isolated point, will show onlyoutliers. X and the two clusters are clearly visible in astemplot of these data.

School of the day!Block 4

School of the day!Block 5

Answers to H.W. E #4 and 5

Answer #5

7.2 Generating Sampling Distributions

Sampling Distribution of a MeanIf you toss a fair die 10,000 what do you expect the histogramof the results to look like?

How would you expect the histogram to look like if you weregraphing the results of the average of two die?

Central Limit Theorem

The sampling distribution of any mean becomes more nearly Normal as the sample size grows. Most importantly the observations need to be independent and collected with randomization.

FYI “central” in the theorem name means “fundamental”

CLT and Equations• The CLT requires essentially the same assumptions andconditions from modeling proportions: Independence,Sample size, Randomization, 10% and large enough sample

The standard deviation of the sampling distribution issometimes called the standard error of the mean.

sizesample/pop.ofdev.standardnx

Properties of Sampling Distribution of Sample Meanpg 430

If a random sample of size n is selected from a populationwith mean and standard deviation , then• The mean of the sampling distribution of x equation

equals the mean of the population, : • The standard deviation , the sampling distribution of , sometimes called the standard error of the mean:

• Using the formula above you can find standard error of the sample mean without simulation

x

x

x

sizesample/pop.ofdev.standardnx

Number of Children Problem

Physical Education Department and BMI studyA college physical education department asked a random sample of 200female students to self-report their heights and weights, but thepercentage of students with body mass indexes over 25 seemedsuspiciously low. One possible explanation may be that the respondents“shaded“ their weights down a bit. The CDC reports that the meanweight of 18-year-old women is 143.74 lb, with a standard deviation of51.54 lb, but these 200 randomly selected women reported a meanweight of only 140 lb.

Question: Based on the Central Limit Theorem and the 68-95-99.7 Rule,does the mean weight in this sample seem exceptionally low or mightthis just be random sample-to-sample variation?

Common Mistakes on Test

Form A

Form B

1. This table gives the percentage of women who ultimately have a given number of children. For example, 19% of women ultimately have 3 children. What is the probability that two randomly selected women will have a combined total of exactly 2 children?

0 and 2, 1 and 1, 2 and 00.18* 0.35 + 0.17*0.17+ 0.35* 0.18 = 0.159

2. For the sake of efficiency, a shoe company decides to produce the left shoe of each pair at one site and the right shoe at a different site. If the two sites produce shoes with a number of defects reflected by ,18.0,005.0and15.0,002.0 2211

what is the mean and standard deviation for the number of defects for pairs of shoes produced by this company.

234.0549.018.015.0

007.0005.0002.022222

YXYX

YXYX

Expected Number of Success and Expected Number of Trials

Binomial Distribution: Flipping a coin 6 times, about howmany times flip head on average.

In a simple random sample of 15 students, how many are expected to be younger than 20 . 60% of students are under 20.

Geometric Distribution: Flipping a coin, when do you expect(on average) to have your first success.

What is the expected number of interview before the second person

without health insurance is found? (16% no health insurance)

3)5.0(6 pnx

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Expected Number of Success and Expected Number of Trials

Binomial Distribution: Flipping a coin 6 times, about howmany times flip head on average.

What is the average number of students without laptops you would expect to find after sampling 5 random students? (60% w/ laptops)

Geometric Distribution: Flipping a coin, when do you expect(on average) to have your first success.

On average how many otters would biologists have to check before finding an infected otter? (20% are infected)

3)5.0(6 pnx

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