In-Class Questions - Type I and Type II Errors
Applet 1
Applet 2
Type I error
A type I error occurs when one rejects the null hypothesis when it is true. The probability of a type I error is the level of significance of the test of hypothesis, and is denoted by *alpha*. Usually a one-tailed test of hypothesis is is used when one talks about type I error.
QUESTION #1:
If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, and men with cholesterol levels over 225 are diagnosed as not healthy, what would be the null and alternative hypothesis for this test? What is the probability that a healthy person will be mistakenly diagnosed as as not healthy? What Type of Error would this misdiagnosis be?
ANSWER:
z=(225-180)/20=2.25; the corresponding tail area is .0122, which is the probability of a type I error.
QUESTION #2:
If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, at what level (in excess of 180) should men be diagnosed as not healthy if you want the probability of a type one error to be 2%?
ANSWER:
2% in the tail corresponds to a z-score of 2.05; 2.05 × 20 = 41; 180 + 41 = 221.
Type II error
A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. The probability of a type II error is denoted by *beta*. One cannot evaluate the probability of a type II error when the alternative hypothesis is of the form µ > 180, but often the alternative hypothesis is a competing hypothesis of the form: the mean of the alternative population is 300 with a standard deviation of 30, in which case one can calculate the probability of a type II error.
QUESTION #3:
If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, but only men with a cholesterol level over 225 are diagnosed as predisposed to heart disease, what is the probability of a type II error (the null hypothesis is that a person is not predisposed to heart disease).
ANSWER:
z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, which is the probability of a type II error (*beta*).
QUESTION #4:
If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, above what cholesterol level should you diagnose men as predisposed to heart disease if you want the probability of a type II error to be 1%? (The null hypothesis is that a person is not predisposed to heart disease.)
ANSWER:
1% in the tail corresponds to a z-score of 2.33 (or -2.33); -2.33 × 30 = -70; 300 - 70 = 230.
Applet 2
Type I error
A type I error occurs when one rejects the null hypothesis when it is true. The probability of a type I error is the level of significance of the test of hypothesis, and is denoted by *alpha*. Usually a one-tailed test of hypothesis is is used when one talks about type I error.
QUESTION #1:
If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, and men with cholesterol levels over 225 are diagnosed as not healthy, what would be the null and alternative hypothesis for this test? What is the probability that a healthy person will be mistakenly diagnosed as as not healthy? What Type of Error would this misdiagnosis be?
ANSWER:
z=(225-180)/20=2.25; the corresponding tail area is .0122, which is the probability of a type I error.
QUESTION #2:
If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, at what level (in excess of 180) should men be diagnosed as not healthy if you want the probability of a type one error to be 2%?
ANSWER:
2% in the tail corresponds to a z-score of 2.05; 2.05 × 20 = 41; 180 + 41 = 221.
Type II error
A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. The probability of a type II error is denoted by *beta*. One cannot evaluate the probability of a type II error when the alternative hypothesis is of the form µ > 180, but often the alternative hypothesis is a competing hypothesis of the form: the mean of the alternative population is 300 with a standard deviation of 30, in which case one can calculate the probability of a type II error.
QUESTION #3:
If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, but only men with a cholesterol level over 225 are diagnosed as predisposed to heart disease, what is the probability of a type II error (the null hypothesis is that a person is not predisposed to heart disease).
ANSWER:
z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, which is the probability of a type II error (*beta*).
QUESTION #4:
If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, above what cholesterol level should you diagnose men as predisposed to heart disease if you want the probability of a type II error to be 1%? (The null hypothesis is that a person is not predisposed to heart disease.)
ANSWER:
1% in the tail corresponds to a z-score of 2.33 (or -2.33); -2.33 × 30 = -70; 300 - 70 = 230.