lane/stat _sim/sampling_dist/index.html

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Example: IQ

• Mean IQ = 100

• Standard deviation = 15

• What is the probability that a person you randomly bump into on the street has an IQ of 110 or higher?

Step 1: Sketch out question

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110

Step 2: Calculate Z score

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110

(110 - 100) / 15 = .66

Step 3: Look up Z score in Table

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110

Z = .66; Column C = .2546

.2546

Example: IQ

• You have a .2546 probability (or a 25.56% chance) of randomly bumping into a person with an IQ over 110.

Now. . . .

• What is the probability that the next 5 people you bump into on the street will have a mean IQ score of 110?

• Notice how this is different!

Population

• You are interested in the average self-esteem in a population of 40 people

• Self-esteem test scores range from 1 to 10.

Population Scores

• 1,1,1,1• 2,2,2,2• 3,3,3,3• 4,4,4,4• 5,5,5,5

• 6,6,6,6• 7,7,7,7• 8,8,8,8• 9,9,9,9• 10,10,10,10

Histogram

012345

678910

1 2 3 4 5 6 7 8 9 10

What is the average self-esteem score of this population?

• Population mean = 5.5

• What if you wanted to estimate this population mean from a sample?

Group Activity

• Randomly select 5 people and find the average score

Group Activity

• Why isn’t the average score the same as the population score?

• When you use a sample there is always some degree of uncertainty!

• We can measure this uncertainty with a sampling distribution of the mean

Characteristics of a Sampling Distribution of the means

• Every sample is drawn randomly from a population

• The sample size (n) is the same for all samples

• The mean is calculated for each sample

• The sample means are arranged into a frequency distribution (or histogram)

• The number of samples is very large

INTERNET EXAMPLE

Sampling Distribution of the Mean

• Notice: The sampling distribution is centered around the population mean!

• Notice: The sampling distribution of the mean looks like a normal curve!– This is true even though the distribution of

scores was NOT a normal distribution

Central Limit Theorem

For any population of scores, regardless of form, the sampling distribution of the means will approach a normal distribution as the number of samples get larger. Furthermore, the sampling distribution of the mean will have a mean equal to and a standard deviation equal to / N

Mean

• The expected value of the mean for a sampling distribution

• E (X) =

Standard Error

• The standard error (i.e., standard deviation) of the sampling distribution

x = / N

Standard Error

• The of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean?

x = / N

Standard Error


x = 15/ 10

Standard Error


4.74 = 15/ 3.16

Standard Error

• The of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50?

4.74 = 15/ 3.16

Standard Error

• The of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50?

4.74 = 15/ 3.16

2.12 = 15/7.07

Standard Error

• The bigger the sample size the smaller the standard error

• Makes sense!

Question

• For an IQ test = 100 = 15

• What is the probability that in a class the average IQ of 54 students will be below 95?

• Note: This is different then the other “z” questions!

Z score for a sample mean

Z = (X - ) / x


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Step 2: Calculate the Standard Error15 / 54 = 2.04

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Step 3: Calculate the Z score(95 - 100) / 2.04 = -2.45

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Step 4: Look up Z score in TableZ = -2.45; Column C =.0071

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.0071

Question

• From a sample of 54 students the probability that their average IQ score is 95 or lower is .0071

lane/stat _sim/sampling_dist/index.html

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