Calculating zscores
In the previous question, we saw that obtaining a sample mean of 550 or greater from a random sample of VASTtest takers is more likely when the sample size is 5 than when the sample size is 25. How likely is each outcome? To find the exact probabilities, we can calculate the zscore associated with a sample mean of 550 with respect to the sampling distributions of means for each sample size and then use the z distribution to find the probability, or pvalue.
The formula to convert a sample mean, X, to a zscore is:
where μ is the population mean, σ is the population standard deviation, and N is the sample size.
From zscores to pvalues
Looking at the numerator (top portion) in the formula, we can see that the more the sample mean deviates from the population mean, the bigger the zscore. Also notice when the denominator (bottom portion, which is the standard error of the mean) is smaller, the obtained zscore is bigger. Bigger or more extreme zscores are associated with smaller pvalues, meaning they are less probable.
With a sample size of N = 25 VASTtest takers, the zscore that corresponds to a sample mean of X = 550 is:
Using the p–z converter, this zscore of 2.5 corresponds to a pvalue of .006. That is, when drawing a random sample of 25 VASTtest takers, the probability of obtaining a sample mean of 550 or greater is 0.6%—a very unlikely event.
What happens with a smaller sample size? How much more likely is it to obtain a sample mean of 550 if we have a sample with N = 5? Answer the next question to find out.
Question D: zscores and Probabilities
VAST scores are normally distributed with a mean score of 500 and a standard deviation of 100. Calculate the zscore and the probability of obtaining a sample mean 550 or greater when N = 5 (assume random sampling; you can compute the zscore and use the p–z converter to convert the zscore into a pvalue).
zscore is: pvalue is:
Show me a hint
Recall that zscores reflect how much a score deviates from the mean. To convert a sample mean Xinto a zscore, subtract the population mean, m, from X and then divide by the standard error of the mean (s divided by the square root of N):
The mean of VAST scores is 500 and the standard deviation is 100. With a sample size of 5, a sample mean of 550 equals a zscore of 1.12. What is the associated probability of this zscore? (You use the p–z converter to find the probability of obtaining a zscore this large or larger.)
Check Answer
With a sample size of 5, the probability of obtaining a sample mean that is 550 or greater is about 13.1% (z = 1.12, p = .131), which is much more likely than when the sample size is 25 (p = .006).
zscore 
Onetailed


A sample mean of 550 or greater when N = 5  .131  
A sample mean of 550 or greater when N = 25  .006 
Notice how the probabilities of 13.1% and 0.6% are related to where 550 falls within the different sampling distributions: