m/sampling_dist/index.html

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 -3  -2  -1   1  2  3  110

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

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

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 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 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/ = 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

Step 2: Calculate the Standard Error 15 / 54 =

Step 3: Calculate the Z score ( ) / 2.04 =

Step 4: Look up Z score in Table Z = -2.45; Column C =

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