The basic facts about hypothesis testing.
Practice problems
In testing for psychic powers, researchers use a deck with five different shapes, as shown in the picture on the left. If the deck is re-shuffled every time, the probability of guessing correctly by pure chance is 1/5 or p = .2 written in decimal. The test would be one tailed high, and we use the z-score table, so the threshold for 95% confidence is = 1.645 and the threshold for 99% confidence is z = 2.325.
Questions:
1. If a subject gets 3 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers?
2. If a subject gets 4 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
3. If a subject gets 5 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
4. If a subject gets 30 out of 100 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
5. If n = 100 and p = .2 in a high one tailed test, find the minimum number of correct answers for rejecting H0 to the 95% confidence level and the 90% confidence level.
Answers in the first comment.
Bonus questions
We have a sample with n = 40, x-bar = 172.5 and sx = 119.5.
1. What is the one tailed low threshold for 95% confidence?
2. What is the one tailed high threshold for 99% confidence?
3. If H0 is mux = 200, are we 95% confident we can reject this for HA: mux < 200?
4. If H0 is mux = 100, are we 99% confident we can reject this for HA: mux > 100?
Answers in second comment.
2 comments:
1. If a subject gets 3 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers?
Answer: z = (.3-.2)/sqrt(.2*.8/10) = .7905..., so we fail to reject H_0.
2. If a subject gets 4 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
Answer: z = (.4-.2)/sqrt(.2*.8/10) = 1.5811..., so we fail to reject H_0 at both the 95% and the 99% confidence level.
3. If a subject gets 5 out of 10 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
Answer: z = (.5-.2)/sqrt(.2*.8/10) = 2.3717..., so we reject H_0 at both the 95% and the 99% confidence level.
4. If a subject gets 30 out of 100 correct in a psychic test, are we 95% confident the subject shows psychic powers? Are we 99% confident?
Answer: z = (.3-.2)/sqrt(.2*.8/100) = 2.5..., so we reject H_0 at both the 95% and the 99% confidence level.
5. If n = 100 and p = .2 in a high one tailed test, find the minimum number of correct answers for rejecting H0 to the 95% confidence level and the 90% confidence level.
Either by trial and error or formula, the first time the z-score gets over 1.645 is with 27 of 100 correct, and the first time it crossed the 2.325 threshold is at 30 of 100.
We have a sample with n = 40, x-bar = 172.5 and sx = 119.5.
1. What is the one tailed low threshold for 95% confidence?
Answer: n=40 means df=39, but we don't have 39, so the closest we can use is n=40. The tail probability for .05 is 1.684, but because this is a low threshold, we negate it and use -1.684. If you use the tinv function in Excel, you get -1.685, a very small difference.
2. What is the one tailed high threshold for 99% confidence?
Answer: n=40 means df=39, but we don't have 39, so the closest we can use is n=40. The tail probability for .05=1 is 2.423. If you use the tinv function in Excel, you get 2.426.
3. If H0 is mux = 200, are we 95% confident we can reject this for HA: mux < 200?
Answer: The test statistic is t = (172.5-200)/119.5*sqrt(40) = -1.455. This is not smaller than our threshold of -1.685, so we fail to reject the null hypothesis that the true average is 200.
4. If H0 is mux = 100, are we 99% confident we can reject this for HA: mux > 100?
Answer: The test statistic is t = (172.5-100)/119.5*sqrt(40) = 3.827. This is much larger than our threshold of 2.423, so we reject the null hypothesis that the true average is 100.
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