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1
probability densityfunction of X
10
This sequence explains the logic behind a one-sided t test.
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
2.5% 2.5%
2
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
We will start by considering the case where can take only two possible values: 0, as under the null hypothesis, and 1, the only possible alternative.
1+2sd
2.5% 2.5%
3
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
An example of this situation is where there are two types of removable lap-top batteries: regular and long life.
1+2sd
2.5% 2.5%
4
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
You have been sent an unmarked shipment and you take a sample and see how long they last. Your null hypothesis is that they are regular batteries.
1+2sd
2.5% 2.5%
5
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
Suppose that the sample outcome is as shown. You would not reject the null hypothesis because the sample estimate lies within the acceptance region for H0.
1+2sd
2.5% 2.5%
6
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
Here you would reject the null hypothesis and conclude that the shipment was of long life batteries.
1+2sd
2.5% 2.5%
7
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
Here you would stay with the null hypothesis.
1+2sd
2.5% 2.5%
8
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd
A sample outcome like this one gives rise to a serious problem. It lies in the rejection region for H0, so our first impulse would be to reject H0.
1+2sd
2.5% 2.5%
9
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
But to reject H0 and go with H1 is nonsensical. Granted, the sample outcome seems to contradict H0, but it contradicts H1 even more strongly.
2.5% 2.5%
10
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
The probability of getting a sample outcome like this one is much smaller under H1 than it is under H0.
2.5% 2.5%
11
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
For this reason the left tail should be eliminated as a rejection region for H0. We should use only the right tail as a rejection region.
2.5%
12
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
The probability of making a Type I error, if H0 happens to be true, is now 2.5%, so the significance level of the test is now 2.5%.
2.5%
13
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
We can convert it back to a 5% significance test by building up the right tail until it contains 5% of the probability under the curve. It starts 1.645 standard deviations from the mean.
5%
14
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
Why would we want to do this? For the answer, we go back to the trade-off between Type I and Type II errors.
5%
15
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
With a 5% test, there is a greater chance of making a Type I error if H0 happens to be true, but there is less risk of making a Type II error if it happens to be false.
5%
16
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
1+2sd0–2sd
Note that the logic for dropping the left tail depended only on 1 being greater than 0. It did not depend on 1 being any specific value.
5%
17
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
1+2sd0–2sd
Hence we can generalize the one-sided test to cover the case where the alternative hypothesis is simply that is greater than 0.
5%
18
probability densityfunction of X
10
ONE-SIDED t TESTS
0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
1+2sd0–2sd
To justify the use of a one-sided test, all we have to do is to rule out, on the basis of economic theory or previous empirical experience, the possibility that is less than 0.
5%
probability densityfunction of X
0 0 +sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : < 0
0+2sd0–2sd
Sometimes, given a null hypothesis H0: = 0, on the basis of economic theory or previous experience, you can rule out the possibility of being greater than 0.
19
ONE-SIDED t TESTS
1
5%
probability densityfunction of X
0 0 +sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : < 0
0+2sd0–2sd
20
ONE-SIDED t TESTS
1
In this situation you would also perform a one-sided test, now with the left tail being used as the rejection region. With this change, the logic is the same as before.
5%
We will next investigate how the use of a one-sided test improves the trade-off between the risks of making Type I and Type II errors.
21
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
2.5% 2.5%
22
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
We will start by returning to the case where can take only two possible values, 0 and 1.
2.5% 2.5%
23
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
Suppose that we use a two-sided 5% significance test. If H0 is true, there is a 5% risk of making a Type I error.
2.5% 2.5%
24
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
However, H0 may be false. In that case the probability of not rejecting it and making a Type II error is given by the blue shaded area.
2.5% 2.5%
25
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
This area gives the probability of the estimate lying within the acceptance region for H0, if H1 is in fact true.
2.5% 2.5%
26
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
Now suppose that you use a one-sided test, taking advantage of the fact that it is irrational to reject H0 if the estimate is in the left tail.
5%
27
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
Having expanded the right tail to 5%, we are still performing a 5% significance test, and the risk of making a Type I error is still 5%, if H0 is true.
5%
28
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
But if H0 is false, the risk of making a Type II error is smaller than before. The probability of an estimate lying in the acceptance region for H0 is now given by the green area.
5%
29
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
This is smaller than the blue area for the two-sided test.
5%
30
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
Thus, with no increase in the probability of making a Type I error (if H0 is true), we have reduced the probability of making a Type II error (if H0 is false).
5%
31
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
When the alternative hypothesis is H1: > 0 or H1: < 0, the more general (and more typical) case, we cannot draw this diagram.
5%
32
ONE-SIDED t TESTS
probability densityfunction of X
10 0+sd0–sd
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 1
0–2sd 1+2sd
Nevertheless we can be sure that, by using a one-sided test, we are reducing the probability of making a Type II error, if H0 happens to be false.
5%
33
One-sided tests are often particularly useful where the analysis relates to the evaluation of treatment and effect. Suppose that a number of units of observation receive some type of treatment and Xi is a measure of the effect of the treatment for observation i.
null hypothesis: H0 : = 0
ONE-SIDED t TESTS
34
To demonstrate that the treatment did have an effect, we set up the null hypothesis H0: = 0 and see if we can reject it, given the sample average X.
null hypothesis: H0 : = 0
ONE-SIDED t TESTS
35
probability densityfunction of X
0
If you use a two-sided 5% significance test, X must be 1.96 standard deviations above or below 0 if you are to reject H0.
null hypothesis: H0 : = 0
alternative hypothesis: H1 : = 0
reject H0reject H0 do not reject H0
1.96 sd-1.96 sd
ONE-SIDED t TESTS
2.5% 2.5%
36
probability densityfunction of X
0
However, if you can justify the use of a one-sided test, for example with H1: > 0, your estimate only has to be 1.65 standard deviations above 0.
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
5%
37
probability densityfunction of X
0
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
This makes it easier to reject H0 and thereby demonstrate that the treatment has had a significant effect.
5%
38
probability densityfunction of X
0
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
Throughout this sequence, it has been assumed that the standard deviation of the distribution of b2 is known, and the normal distribution has been used in the diagrams.
5%
39
probability densityfunction of X
0
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
In practice, of course, the standard deviation has to be estimated as the standard error, and the t distribution is the relevant distribution. However, the logic is exactly the same.
5%
40
probability densityfunction of X
0
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
At any given significance level, the critical value of t for a one-sided test is lower than that for a two-sided test.
5%
41
probability densityfunction of X
0
reject H0do not reject H0
1.65 sd
ONE-SIDED t TESTS
null hypothesis: H0 : = 0
alternative hypothesis: H1 : > 0
Hence, if H0 is false, the risk of not rejecting it, thereby making a Type II error, is smaller.
5%
Copyright Christopher Dougherty 2011.
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