. If the data is in the form of counts, you will be using a count (X) or an observed proportion (
).Significance testing plays a central role in chapters 6, 7, and 8. Although the procedure for doing a significance test is pretty much the same, the details and especially the formulas to use can vary widely. Here is a guide to determining which formulas and tables to use to conduct a significance test.
Data from an experiment or study falls into two broad categories. If the data is numerical, you will be measuring . If the data is in the form of counts, you will be using a count (X) or an observed proportion ().
For numerical data, the relevant question to ask is whether or not you know the standard deviation for the underlying random process. If you know 
, you will use the normal distribution. If you don't know 
ahead of time, you will use the sample standard deviation s to estimate 
and you will have to use the t distribution.
For counts, if the sample size n is less than or equal to 20 and you know p ahead of time, you should use the binomial distribution. If you don't know p ahead of time, you will have to use the observed proportion 
to estimate it. Also, if 
is close to 0 or 1 and n is not too large, you may want to replace 
with the Wilson estimate, (X+2)/(n + 4). If n is greater than about 15, you can safely use the normal distribution to approximate the binomial distribution.
If you are looking at the difference of two samples, the null hypothesis usually says that there is no difference in the two samples. If you are looking at a single sample, the problem statement will tell you what to use for the null hypothesis.
There are about a half-dozen different formulas for the standard error, but you can quickly determine which formula to use. The first question to ask is whether you are looking at a single sample or a difference of two samples. The second question is whether you are measuring 
or 
. The third question is whether you know 
or p ahead of time or have to estimate them by s or 
.
These are the formulas to use for the standard error in a single sample.
 known? | Numerical | Proportions |
|---|---|---|
| Yes |  |  |
| No |  |  |
These are the formulas to use for the difference of two samples. The proportions formula in this case uses the pooled estimate, 
 known? | Numerical | Proportions |
|---|---|---|
| Yes | N. A. | N. A. |
| No |  |  |
Next, you are going to compute a one or two sample z-score or t-score.
Note that in almost every case where you are comparing two samples, the null hypothesis says that 
or 
.
The final step is to compute P from the relevant distribution and compare it to the significance level you are trying to beat.
In computing P, you have to pay attention to the one-sided/two-sided alternative distinction. The one-sided alternative applies in situations where the alternative hypothesis says that one value is to one side of another. If you don't know in advance which side an alternative will fall on and you simply want to express the fact that two things are different, you should use the two-sided alternative. The practical difference is that in the two-sided alternative you double P after computing it. That makes it comparatively more difficult to reject the null hypothesis.
Here is a list of other concepts I may test you on. The form of the questions I may ask will vary. In some cases I will ask for definitions or a brief discussion, while in other cases you will have to compute something.
Distribution
Mean
Median
Variance
Standard Deviation
Stemplot
Histogram
Outlier
Box plot
1.5 IQR range test
Explanatory variable, response variable
Scatterplot
Positive and negative association
Correlation
Lurking variable
Regression line
r2
Experiment, Block design
Matched pairs experiment
Bias
Confidence interval
Joint distribution
Marginal distribution
Conditional distribution
test
These are the formulas I will give you on the final exam. If a formula does not appear here either you will not need it on the exam or I think it is sufficiently simple that you can just memorize it.
Here are four sample questions covering material found in chapters 6, 7, 8, and 9. Solutions are available here.
1. Eighth grade students in a certain state are required to take a standardized acheivement test in mathematics. The state-wide mean score on the exam is 760 out of a possible 1000 with a standard deviation of 70. The mean score at a particular school with 150 eighth graders is 800. Is this significantly greater than state-wide mean?
2. The 1958 Detroit Area Study was an important sociological investigation of the influence of religion on everyday life. The sample was basically a simple random sample of the population of the metropolitan area. Of the 656 respondents, 267 were white Protestants and 230 were white Catholics. One question asked whether the government was doing enough in areas such as housing, unemployment, and education; 161 of the Protestants and 136 of the Catholics said No. Is there evidence that white Protestants and white Catholics differed on this issue?
3. In January 1975, the Committee on Drugs of the American Academy of Pediatrics recommended that tetracycline drugs not be given to children under the age of 8. A two-year study conducted in Tennessee investigated the extent to which physicians had prescribed these drugs between 1973 and 1975. The study categorized family practice physicians according to whether the county of their practice was urban, intermediate, or rural. The researchers examined how many doctors in each of these categories prescribed tetracycline to at least one patient under the age of 8. Here is the table of observed counts
| Urban | Intermediate | Rural | |
|---|---|---|---|
| Tetracycline | 65 | 90 | 172 |
| No tetracycline | 149 | 136 | 158 |
Carry out a significance test to determine whether county type and tetracycline use are related.
4. A bank compares two proposals to increase the amount that its credit card customers charge on their cards. Proposal A offers to eliminate the annual fee for customers who charge $2400 or more during the year. Proposal B offers a small percent of the total amount charged as a cash rebate at the end of the year. The bank offers each proposal to an SRS of 150 of its existing credit card customers. At the end of the year, the total amount charged by each customer is recorded. Here are the summary statistics:
| n |  | s | |
|---|---|---|---|
| A | 150 | $1987 | $392 |
| B | 150 | $2056 | $413 |
Is there a significant difference between the amounts charged by the two groups?