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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 -3 -2 -1  1 2  3 

Step 1: Sketch out question 110 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (110 - 100) / 15 = .66 110 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = .66; Column C = .2546 110 .2546 -3 -2 -1  1 2  3 

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

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

EXCEL

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

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 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 = 15/ 10

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? 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

Step 1: Sketch out question -3 -2 -1 0 1 2 3

Step 2: Calculate the Standard Error 15 / 54 = 2.04 -3 -2 -1 0 1 2 3

Step 3: Calculate the Z score (95 - 100) / 2.04 = -2.45 -3 -2 -1 0 1 2 3

Step 4: Look up Z score in Table Z = -2.45; Column C =.0071 .0071 -3 -2 -1 0 1 2 3

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