Make a Diagram 2/27/08
Make a Diagram
Means
Interval for / Confidence Interval / Hypotheses / Test Ratio / Critical ValueMean /
or
/ / or
/ or
Difference
between Two
Means /
(See table 3 for
and )
/ / or
/
or
If or ( is usually zero), we have a two-sided test. If or , we have a left-sided test. If or , we have a right-sided test.
Make a diagram. This diagram could look like a Normal curve, even when the distribution is not symmetrical.
Confidence Interval: The idea is to draw the confidence interval and to reject the null hypothesis
for a test of one mean if is not on the confidence interval
or for a test of two means if (usually zero) is not on the confidence interval.
Whether the confidence interval is one or two-sided, it will include for a test of one mean and for a test of two means. or will be in the center of your diagram.
This means that for a left-sided test the confidence interval will be made by replacing ‘=’ by ‘’ and ‘’ by ‘+.’ The interval will start somewhat to the right of the center of the diagram and include the far left corner of the diagram.
This also means that for a right-sided test the confidence interval will be made by replacing ‘=’ by ‘’ and ‘’ by ‘–.’ The interval will start somewhat to the left of the center of the diagram and include the far right corner of the diagram.
Test Ratio: A diagram will always have zero at its center. Zero will never be in the ‘reject zone. The test ratio will be compared to , , , , , as appropriate. Let us call your calculated value of the ratio or .
For a left-sided test the ‘reject’ zone will be to the left of zero and will go from or to the far left corner of the diagram. Reject the null hypothesis if or is in the ‘reject’ zone.
For a right-sided test the ‘reject’ zone will be to the right of zero and will go from or to the far right corner of the diagram. Reject the null hypothesis if or is in the ‘reject’ zone.
P-value: The p-value is defined as the probability of getting results as extreme or more extreme than those actually observed. It is usually calculated by getting a test ratio. Let us call your calculated value of the ratio or .
For a left-sided test, or .
For a right-sided test, or .
For a 2-sided test, you can take the smaller of and or the smaller of and .
To do a conventional hypothesis test, reject the null hypothesis if the p-value is below the significance level.
Critical value for the sample mean or the difference between two sample means: The critical value is used to find a ‘reject’ zone. The null hypothesis mean, , or the null hypothesis difference between two means, , is never in the reject zone and is always in the center of the diagram. The critical value is compared to or .
This means that for a left-sided test the ‘reject’ zone will be to the left of (the mean from the null hypothesis) or and will be found by replacing and ‘’ by ‘–.’ The ‘reject’ zone will start somewhat to the left of the center of the diagram and include the far left corner of the diagram.
This also means that for a right-sided test the ‘reject’ zone will be to the right of (the mean from the null hypothesis) and will be found by replacing and ‘’ by ‘+.’ The ‘reject’ zone will start somewhat to the right of the center of the diagram and include the far right corner of the diagram.
Proportions
Interval for / Confidence Interval / Hypotheses / Test Ratio / Critical ValueProportion /
/ / /
Difference
between
proportions
/
/
/
Or use /
If you replace with or with and always use and never , the discussion of means should explain this pretty well.
Variances
Variance-
Small Sample / / / /
Variance-
Large Sample / / / /
Ratio of Variances
/
/ / and
Confidence Intervals: The confidence interval formulas for two-sided intervals are explained in Confidence Limits and Hypothesis Tests for Variances and the outline documents extracted from them.
For actual hypothesis tests, the only method commonly used is a Test Ratio. For a small sample you are using a diagram of the statistic with degrees of freedom. The center of the diagram will be near the mean of which is equal to the number of degrees of freedom. This will never be in a ‘reject’ zone.
For a single variance, just as above, or is a two sided test. The lower ‘reject’ zone will be below . (For example, if and this will be below the .975 value in the part of the chi-square table. ) The upper ‘reject’ zone will be above . (For example, if and this will be above the .025 value in the part of the chi-square table. ) If your computed value of falls in one of the ‘reject’ zone, reject the null hypothesis.
For a single variance, just as above, or is a left-sided test. The ‘reject’ zone will be below . (For example, if and this will be below the .95 value in the part of the chi-square table.) If your computed value of falls in the ‘reject’ zone, reject the null hypothesis.
For a single variance, just as above, or is a right-sided test. The ‘reject’ zone will be above . (For example, if and this will be above the .05 value in the part of the chi-square table.) If your computed value of falls in the ‘reject’ zone, reject the null hypothesis.
For a large sample, compute and make it into a value of . Then treat it like a test ratio for the mean or proportion.
For a ratio of variances, because the F table is set up for right-sided tests, all tests should be done as right-sided tests. If we have or , we have a true right-sided test. Test the ratio against the critical value . Your diagram should show a center at 1 and a ‘reject’ zone above (For example, if , and this will be above the .05 value in the , part of the F table.) Reject the null hypothesis if your value of falls in the ‘reject’ zone.
If we have or , we have a left-sided test. Think of it as a right-sided test with or . Test the ratio against the critical value . Your diagram should show a center at 1 and a ‘reject’ zone above (For example, if , and this will be above the .05 value in the , part of the F table.) Reject the null hypothesis if your value of falls in the ‘reject’ zone.
If you have to do a two-sided test, think of it as two one-sided tests. Test the ratio against the critical value (for example ). Then test the ratio against the critical value for example ). If either of these two tests results in a rejection, reject the null hypothesis. In practice, only one of these tests actually needs to be done, since if is above one, will be below 1 and critical values from the F table cannot be below 1.
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