Let's talk about a different scenario in which the outcome is dichotomous, that is, a "yes or no" outcome, and we compare one sample to the known population parameter, "p" which is the population proportion. Examples might be whether surgery was successful (yes/no), cancer remission (yes/no). We still have one study sample, and what we measure on each participant is whether they had the outcome (yes or no).
The data we observe from out sample are the sample size, "n", and the number of positive responses, which we convert into a sample proportion by dividing the number of positive responses by the sample size.
There is only one test statistic formula for this type of test, and it is a Z formula.
This is appropriate as long as one condition is met, shown at the bottom. That is that the smaller of the sample proportion and one minus the sample is at least 5, where p zero is the population parameter specified in the null hypothesis.
Again, the null hypothesis is that the sample proportion (p hat) is equal to the population proportion, p sub zero.
The alternative hypothesis can be one of 3 options: that the sample proportion is greater than p sub zero, it's less, or it's just different from p sub zero.
An example: the National Center of Health Statistics reports the prevalence of cigarette smoking in adults was 21.1% in 2002. Is the proportion of smokers lower in the
Framingham cohort?
Suppose that in 3536 participants in Framingham, 482 reported cigarette smoking. 482 divided by 3536 = 0.136 or 13.6%. Is 13.6% statistically significantly lower than the proportion reported by NCHS? The null hypothesis is that the proportion of smokers in Framingham is the same as the national proportion, which was 0.211, or 21%. And we write the null and alternative hypotheses as proportions, not as percentages. And we are hypothesizing that the proportion of smokers in Framingham is lower than 0.211, so this is a lower, one-sided test.
The test statistic is the Z statistic, and there is no problem with the sample size here.
The decision rule comes from the Z-table. We will reject the null hypothesis if Z is less than or equal to minus 1.645. We then substitute our data into the formula. We subtract the population proportion from the sample proportion and divide by the standard error, which is the square root of p0 time (1 minus p0) divided by n. We get Z equal to minus 10.93, which is a huge negative Z value. Negative 10.93 is off the chart on the left-hand side.
Our conclusion is to reject the null hypothesis (H0). We have statistically significant evidence that the proportion of cigarette smokers in the Framingham cohort is lower than the national proportion. And the p-value here would be very small, less than 0.0001.
Excel doesn't have a procedure for one-sided tests, but one could program the calculation of Z into Excel and then use the formulas mention in the previous video to calculate the one-sided p-value.