Fundamentals of Statistics 3: Sampling :: 1-Sample Z-test
Statistics is all about understanding the role of chance in our measurements and we often want to know what the chances are of obtaining a sample mean given the population mean is a certain value. The standard error of the mean tells us how much the sample mean varies from sample to sample (it is the standard deviation of the population mean given a particular sample size). The empirical rule tells us that 95% of the time the sample mean will fall within two standard errors of the population mean. We can extend the principle of the empirical rule and use the normal curve to find the probabilities for a given sample mean using a statistical test called the 1-sample z-test.

We saw how a z-score can show us the probability of obtaining a single value, say an IQ score or height, if we know the population mean and standard deviation. Now we'll look at the probability of obtaining a sample mean, given we know the population mean and standard deviation (which we almost never do, but bear with me).  It might sound confusing but the difference is between finding the probability of individual men's heights being above 6'2, versus finding the probability a sample of men's heights is above 6'2. We just need to update our z-score formula with the standard error of the mean instead of the standard deviation.

 Z-Score for a Single Value Z-Score for a Sample Mean (1-Sample Z Test)

The blue parts of the z-score for a sample mean formula show the slight change from the z-score formula. We're using the sample mean instead of one-point (that's the bar over the x) and we're taking into account the sample size as well. So the z-score for a sample mean formula will tell us how many standard errors there are between the sample mean and the population mean. If the sample mean was identical to the population mean, the numerator would be 0, making the equation 0 (meaning 0 standard errors from the mean).

For example, we wondered how likely it would be to find a random sample of 30 women with an average height of 6'1 (73 inches) knowing the general population of women have a mean of 65 inches and a standard deviation of 3.5 inches. To find out we insert this value in the 1-sample Z test formula.

First let's find the standard error of the mean = 3.5/SQRT(30) = .639. So at a sample size of 30, we would expect 68% of the sample means to fluctuate by about half an inch, or between 65 -.693 and 65+.693 = 64.4 and 65.6 inches. A sample of 30 women with a mean height of 71 inches would then be (73-65)/ .639 = 12.52 standard errors above the mean. Since we know 99% of the sample means would fall within 3 standard errors (from the empirical rule), the probability of this sample mean is less than 1%. The actual probability can be looked up using a normal table or an online calculator, although most don't go this high.

So what do we conclude?  Well, we said it would be unusual (like winning the lottery) and so we conclude that this sample is extremely unlikely (less than .0001% in fact) to have come from the general population of women. We would conclude it is likely from another population of women, for example Olympic Volleyball players.

### Probability of a Single Value versus a Sample Mean

Now it would certainly seem more plausible to find at least one woman from the general population who is 73 inches. Using the z-score formula for a single value, we get a z-score of (73-65)/3.5 = 2.28. Looking up this z-score in a normal table we would expect to encounter a single woman this tall about 1.1% of the time.  While that is not likely, it is more plausible than the .0001% probability of a sample mean of 30 women being 73 inches.

This is a concept that statistics professors like to ask about. We can see that it is more likely to obtain an extreme raw value
than an extreme sample mean. What's more, there is less of a chance of obtaining an extreme sample mean for a large sample size than a small sample size.

If you need more practice understanding z-scores, I put together a Crash Course in Z-scores which provides a visual and easy-to-follow guide with plenty of practice understanding the normal curve and z-scores.

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