**Hypothesis testing: An example using height **

Research question: Are UO men taller than men in 19^{th} century England?

Galton, who loved to measure things, measured vast quantities of English men and determined
that their mean height was 67 inches. The standard deviation is 3, and thus the variance is 9.

1. Restate research question as two hypotheses: H_{A} & H_{0}

(our research hypothesis and our null hypothesis)

H_{A} : UO men are taller than the 19th Englishmen [directional hypothesis]

H_{0}: UO men are the same height as the 19^{th} Century Englishmen [no difference, or shorter]

To put this another way:

H_{A} : The MEAN for UO men is higher than the MEAN for 19th Englishmen mu-UO > mu-Brits

H_{0}: The means are the same.... no difference... (or UO men are shorter) : mu-UO mu-Brits

NOTICE: We are not trying to determine if our sample mean is higher than 67. We know that as
soon as we calculate the mean for our sample. What we are trying to determine is if the height of
UO men (the population of interest) is significantly higher than 67, based on what we have
learned from our sample.

2. Determine sampling distribution (the comparison distribution)

In this case, it will be a sampling distribution of MEANS. But it has to be a distribution of
means from the population we know (19^{th} century men) because we don't KNOW the parameters
for the population we are REALLY interested in (UO men)....

Our N = 18 Large enough?

Answer: Yes, because we know that height is normally distributed variable.

The mean for sampling distribution will be 67.

The variance of sampling distribution will be sigma^{2}/n. 9/18 = .5

The standard deviation (the standard error of the mean) will be the square root of sigma^{2}/n.

Square root of .5 = .71 [notice that when you take the square root of a fraction it gets BIGGER]

So our standard error (SE) = .71

The shape of the sampling distribution will be normal (because the underlying population of
scores is normal, even though our *n* is a little small).

3. Find critical value (on the sampling distribution)

This is what problems in Homework #4 were about. Let's pick a level of .05 (a standard cutoff for psychology research). So now we need to find what that value would be.

** Draw normal curve***

Will the shaded area be in one tail or two?

Answer: One, since we have a DIRECTIONAL hypothesis, and have decided it is actually non-plausible that the mean height of UO men is SHORTER than 19^{th} century Englishmen.

Find the Z score for 5% (in normal curve table, in the back of your book). +1.65. Just have one
value since we have one tail.

4. Find value for sample mean (calculate statistic)

Done. x-bar-UO = 72.08. Z score of (72.08-67)/SE = 5.08/.71 = Z of 7.15

5. Compare 3 & 4 and make decision about H_{0 }: Reject or retain? How do we answer the
research question?

Is the x-bar -UO larger than the critical value, or not? Answer: DEFINITELY LARGER!

Because it is larger in the right direction (in the shaded region) we reject the null hypothesis.
We conclude, with 95% confidence, that UO men are taller. 95% confidence means the
probability that we are wrong about this is less than 5%,* p* < .05.

If it had been smaller (not in the shaded region), we would retain the null hypothesis (fail to
reject). We would say the difference is not significant at the .05 level.

What's the answer to our research question? We conclude that UO men ARE taller than 19^{th}
century Englishmen, at 95% confidence level. Actually, since our value was SO extreme, we
have exceeded the 99% confidence level as well. The general practice is to report that your
result exceeds a more stringent level if that's what you find: you will encounter *p* < .01, or *p* <
.005, or *p* < .001 in reports of results, even though psychologists hardly ever actually choose *p*
< .001 as their cutoff level.

Reasons for caution: We shouldn't be TOO certain, however, because we did violate some assumptions of inferential statistics. We didn't take a random sample of UO men. Perhaps men who take 302 are, for whatever reason, more likely to be tall. Maybe basketball players like statistics? Are more likely to be pre-psych majors? To be REALLY sure, we would want to take another sample, preferably larger, preferably a better approximation of random sampling.