For healthy individuals the level of prothrombin in the blood is approximately n
ID: 3220818 • Letter: F
Question
For healthy individuals the level of prothrombin in the blood is approximately normally distributed with mean 20 mg/100 mL and standard deviation 4 mg/100 mL. Low levels indicate low clotting ability. In studying the effect of gallstones on prothrombin, the level of each patient in a sample is measured to see if there is a deficiency. Let mu be the true average level of prothrombin for gallstone patients. (a) What are the appropriate null and alternative hypotheses? (b) Let X^bar denote the sample average level of prothrombin in a sample of n = 20 randomly selected gallstone patients. Consider the test procedure with test statistic X^bar and rejection region x^bar lessthanorequalto 17.92. What is the probability distribution of the test statistic when H_0 is true? What is the probability of a Type I error for the test procedure? (c) What is the probability distribution of the test statistic when mu = 16.7? Using the test procedure of part (b), what is the probability that gallstone patients will be judged not deficient in prothrombin, when in fact mu = 16.7 (a Type II error)? (d) Consider the standardized test statistic Z = X^bar - 20/sigma/Squareroot n = X^bar - 20/0.8944. What are the values of Z corresponding to the rejection region of part (b)?Explanation / Answer
Given,
Mean(µ)=20mg/100ml
Standard Deviation(SD)=4mg/100ml
A.a) Null Hypothesis H0: µ>20
Alternate Hypothesis H1: µ20
A.b) sample size(n)=20
Mean(µ)=20mg/100ml
Standard Deviation(SD)=4mg/100ml
sample mean(X)=17.92
Null Hypothesis H0: X>17.92
Alternate Hypothesis H1: X17.92
test statistic(Z)=(17.92-20)/4/sqrt(20)
solving we get, Z=-2.32
the appropriate probability of type 1 error is 0.01 or 1%
A.c) sample size(n)=20
Mean(µ)=16.7mg/100ml
Standard Deviation(SD)=4mg/100ml
sample mean(X)=17.92
Null Hypothesis H0: X>17.92
Alternate Hypothesis H1: X17.92
test statistic(Z)=(17.92-16.7)/4/sqrt(20)
solving we get, Z=1.36
the appropriate probability of type 2 error is 1-0.91=0.09 or 9%
A.d) the values of Z corresponding to rejection region are -2.32 to infinity
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