Math 251 Science Supplement for Exam 5
1. In a random sample of 15 patients from the population of all potential patients the mean serum-creatinine level 24
hours after receiving a newly proposed antibiotic was 1. 4 mg/dL. Suppose that the standard deviation in the entire
potential treated population is known to be 0. 5 mg/dL. Suppose the mean serum-creatinine level in the general
population is 1. 0 mg/dL with a standard deviation of 0. 5 mg/dL. Assume that serum-creatinine level is normally
distributed.
(a) Formally state the appropriate hypotheses neccessary to determine whether the mean serum-creatinine
level of the potential treated population is different than the mean serum-creatinine level of the general
population.
(b) Test the hypotheses in part (a) above at the 0.02 level of significance using the critical value method.
(c) Find the approximate p-value for the test.
(d) Compute a two-sided 98% confidence interval for the mean serum-creatinine level in the potential treated
population.
(e) What is the relationship between the answers in parts (b), (c), and (d) above?
2. In a random sample of 21 people from the population of all sedentary people ages 35-44 years the mean plasmaglucose concentration is 5.2 mg/dL with a standard deviation of 0.4 mg/dL. Suppose the mean plasma-glucose
concentration in the general population ages 35-44 years is 4.8 mg/dL with a standard deviation of 0. 5 mg/dL.
Assume that plasma-glucose concentration is normally distributed.
(a) Formally state the appropriate hypotheses neccessary to determine whether the mean plasma-glucose
concentration in the sedentary population is different than the mean plasma-glucose concentration in the general
population.
(b) Test the hypotheses in part (a) above at the 0.01 level of significance using the critical value method.
(c) Find the approximate p-value for the test.
(d) Compute a two-sided 99% confidence interval for the mean plasma-glucose concentration in the
sedentary population.
(e) What is the relationship between the answers in parts (b), (c), and (d) above?
(f) Formally state the appropriate hypotheses neccessary to determine whether the variance of the plasmaglucose concentration in the sedentary population is different than the variance of the plasma-glucose
concentration in the general population.
(g) Test the hypotheses in part (f) above at the 0.01 level of significance using the critical value method.
(h) Find the approximate p-value for the test.
(i) Compute a two-sided 99% confidence interval for the variance of the plasma-glucose concentration in
the sedentary population.
(j) What is the relationship between the answers in parts (g), (h), and (i) above?
3. Suppose that a new drug is tested on 50 patients and 40 of the 50 patients were cured by the new drug. Suppose
it is known that 30% of patients would improve with a placebo (fake treatment).
(a) Formually state the appropriate hypotheses necessary to determine whether the success rate of the new drug is
different than the success rate of a placebo.
(b) Test the hypotheses in part (a) above at the 0.01 level of significance using the critical value method.
(c) Find the approximate p-value for the test.
(d) Compute a two-sided 99% confidence interval for the proportion of people in the population of all patients who
would be cured by the new drug.
(e) What is the relationship between the answers to parts (b), (c), and (d) above?
(f) Formually state the appropriate hypotheses necessary to determine whether the success rate of the new drug is
higher than the success rate of a placebo.
(g) Test the hypotheses in part (a) above at the 0.01 level of significance using the critical value method.
(h) Find the approximate p-value for the test.
(i) Compute an appropriate one-sided 99% confidence interval for the proportion of people in the population of all
patients who would be cured by the new drug.
(j) What is the relationship between the answers to parts (g), (h), and (i) above?
Math 251 Science Supplement to Exam 5
page two
4. Show that the three methods for testing hypotheses about the mean when the standard deviation is known are
equivalent.
Math 251 Partial Answers to Science Supplement for Exam 5
1.
(b)  2.327  3.098  2.327 ? No. Reject Ho.
(c) 0.002
(d) (1.1, 1.7) mg/dL
2.
(b)  2.845  4.583  2.845 ? No. Reject Ho
(c) p < 0.01
(d) (4.95, 5.45) mg/dL
(g) 7.434  12.8  39.997 ? Yes. Fail to reject Ho
(h) 0.24
(i) (0.08, 0.43)
3.
(b)  2.575  7.715  2.575 ? No. Reject Ho
(g) 7.715  2.327 ? No. Reject Ho
(c) p < 0.0002
(h) p < 0.0001
(d) ( 0.63, 0.97 )
(i) ( 0.65, 1 )
Math 251 Partial Answers to Science Supplement for Exam 5
4. We want to test
Ho :   o
versus
H1 :   o
when the population standard deviation  is known.
Using the confidence interval method we will fail to reject Ho
if and only if o is in the confidence interval
if and only if
x  z1 / 2 
if and only if
 z1 / 2 
if and only if
z1 / 2 
if and only if
 z1 / 2 
if and only if
 z1 / 2 
x  z1 / 2 
n , x  z1 / 2 
n  o  x  z1 / 2 
n  o  x  z1 / 2 
n
n
n
(Why?)
n  x  o   z1 / 2 
n
(Why?)
n  x  o  z1 / 2 
n
(Why?)
 x  o 

n
 z1 / 2
(Why?)
Thus the confidence interval method is equivalent to the critical value method.
The above set of inequalities is true
if and only if
if and only if
if and only if
if and only if
if and only if
if and only if
x  o n

 z1 / 2

x  o n 
  Pr z  z1 / 2 
Pr z 




x  o n 
  1   / 2
Pr z 




x  o n 
  1   / 2
 Pr z 




x  o n 
   / 2
1  Pr z 





x  o n  
   
21  Pr z 





(Why?)
(Why?)
(Why?)
(Why?)
(Why?)
if and only if p-value  .
Thus the critical value method is equivalent to the p-value method and thus all three methods are equivalent.