When a person is well, the test result for a certain disease is nor mally distri

Question

When a person is well, the test result for a certain disease is nor mally distributed with mu = 10 (null hypothesis) and sigma = 2. For individuals with the disease, the mean test result is also normally distributed with mu = 15 (alternative hypothesis) and sigma = 2. We classify a person as not having the disease if the test result does n ot exceed c and having the disease if the test result exceeds c . Of course, this classification produces errors: classifying a non - diseased person as having the disease and a person that has the disease as not having the disease.

a) What must c have to be if alpha = .05?

b. Based on the value of c in part a) what is the probability of a Type II error?

c. What would be the P - value for a test result of 14?

Explanation / Answer

a) What must c have to be if alpha = .05?

Null Hypothesis : H0 : mu = 10

Alternative Hypothesis : Ha : mu = 15

for one tailed test at mu (null hypothesis)

c =< mu + 1.645 * sigma = 10 + 1.645 *2 = 13.29

c have to be 13.29.

b. Probability of type II error?

Type II error mean accepting the null hypothesis if it is false.

Pr(X<=13.29; 15;2) that means probability of test results equal or less than 13.29 when he is really ill from the disease.

Z = ( 13.29 -15)/2 = -0.855

P - value for the given Z = 0.1963

so probability of type II error = 0.1963

c. P - value for a test result of 14

Pr (X>= 14; 10;2)

Z = 2 so P - value = 1-0.9772 = 0.0228

P - value = 0.0228

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