1. From Last week

2. Q22
A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers
What is the probability of guessing correctly for any question?

3. Q22
A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers
On average, how many questions would a student get correct for the entire test?

4. Q22
A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers
What is the probability that a student would get more than 15 answers correct simply by guessing?

5. Q22
A multiple choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers
What is the probability that a student would get 15 or more answers correct simply by guessing?

6. The Distribution of Sample Means

7. Review
z-scores close to zero indicate that the sample mean is relatively close to the population mean
z-scores beyond 2 represent extreme values that are quite different from the population mean

8. The Distribution of Sample Means
Definition: the set of means from all the possible random samples of a specific size (n) selected from a specific population
Theoretical distribution because normally you wouldn’t have all the information about a population available, and that’s why we have to use samples to make inferences about the population
Theoretically, we can take every possible sample of a given size from a population, compute all the sample means, and then form a distribution of sample means.
Distribution of sample means: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. The fact that these samples are random is important. There would be sampling with replacement, and each individual within the population could end up in more than one sample.
But I’m calling the distribution of sample means theoretical, because normally you wouldn’t have all the information about a population available, and that’s why we have to use samples to make inferences about the population. It’s usually the whole population that we want to describe, and not just the subset of the population that’s included in our sample, but often it’s impractical or even impossible to actually measure each individual within the population of interest.
So then, using this theoretical distribution, we can determine how close a particular mean is to the population mean, and calculate the probability of obtaining a sample with any particular mean. If there were exactly 100 possible samples that could be obtained from a given population, then the probability of obtaining any one in particular would be 1 out of 100, or .01.
This distribution has well-defined (and predictable) characteristics that are specified in the Central Limit Theorem

9. Characteristics of the distribution of sample mean
The sample means tend to cluster around the population mean
The distribution of sample means can be used to answer probability questions about sample means
The distribution of sample means is approximately normal in shape

10. Central Limit Theorem
1. The mean of the distribution of sample means is called the Expected Value of M
2. The standard deviation of the distribution of sample means is called the Standard Error of M
3. The shape of the distribution of sample means tends to be normal.

11. Example
Imagine a population that is normally distributed with µ=110 and σ=24
If we take a sample from this population, how accurately would the sample mean represent the population mean?

12. q6
For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes
N=4 scores

13. q6
For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes
N=16 scores

14. q6
For a population with a mean of µ=70 and a σ=20, how much error, on average, would you expect between the sample mean and the population mean for each of the following sample sizes
N=25 scores

15. q8
If the population standard deviation is 8, how large a sample is necessary to have a standard error that is?
Less than 4 points?

16. q8
If the population standard deviation is 8, how large a sample is necessary to have a standard error that is?
Less than 2 points?

17. q8
If the population standard deviation is 8, how large a sample is necessary to have a standard error that is?
Less than 1 point?

18. Probability and Sample Means
Because the distribution of sample means tends to be normal, the z-score value obtained for a sample mean can be used with the unit normal table to obtain probabilities.

19. z-Scores and Location within the Distribution of Sample Means
Within the distribution of sample means, the location of each sample mean can be specified by a z-score,
M – μ
z = ───── σM

20. Example
GRE quantitative scores are considered to be normally distributed with a  = 500 and  = 100.
An exceptional group of 16 graduate school applicants had a mean GRE quantitative score of 710.
What is the probability of randomly selecting 16 graduate school applicants with an even greater mean GRE quantitative score?
p (> 710 ) = ?

21. q11
A sample of n=4 scores has a mean of m=75. Find the z-score for this sample
If it was obtained from a population with  = 80 and  = 10

22. q11
A sample of n=4 scores has a mean of m=75. Find the z-score for this sample
If it was obtained from a population with  = 80 and  =40

23. q14
The population IQ scores forms a normal distribution, with a mean of 100 and standard deviation of 15. What is the probability of obtaining a sample mean greater than M=97
For a random sample of n=9 people?

24. q14
The population IQ scores forms a normal distribution, with a mean of 100 and standard deviation of 15. What is the probability of obtaining a sample mean greater than M=97
For a random sample of n=25 people?

25. Example 2
Scores on a test form a normal distribution with µ=70 and σ=12. With a sample size of n=16.
What is the probability of obtaining a sample of at least 75? 70 75

26. What proportion of the sample means will be lower than 73?
Proportion in body

27. What is the probability of obtaining a sample with a mean less than 64?

28. What proportion of the sample means will be within 2 points of the population mean?

29. Example 3
A normal distribution with µ=80 and σ=15, sample size of n=36.
What sample means would mark off the most extreme 5% of the distribution?
What should you do?

30. q17
A population of scores forms a normal distribution with a mean of µ=80 and σ=10
What proportion of the scores have values between 75 and 85?

31. q17
A population of scores forms a normal distribution with a mean of µ=80 and σ=10
For samples of n=4, what proportion of the samples will have means between 75 and 85?

32. q17
A population of scores forms a normal distribution with a mean of µ=80 and σ=10
For samples of n=16, what proportion of the samples will have means between 75 and 85?