Question

Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken twelve blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.85 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (Round your answers to two decimal places.)

lower limit ____

upper limit ____

margin of error ____

(b) What conditions are necessary for your calculations? (Select all that apply.) uniform distribution of uric acid σ is known σ is unknown n is large normal distribution of uric acid

(c) Interpret your results in the context of this problem.

a-The probability that this interval contains the true average uric acid level for this patient is 0.05.

b-There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.

c-There is not enough information to make an interpretation.

d-The probability that this interval contains the true average uric acid level for this patient is 0.95.

e-There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.16 for the mean concentration of uric acid in this patient's blood. (Round your answer up to the nearest whole number.) _____ blood tests

Answer 1

(a) The 95% confidence interval for mean is

Note Though population standard deviation is given , we use t distribution as sample size is small, and we use in place of s

degrees of freedom = n-1 = 12-1=11

For 95% confidence ( =0.05) with df =11 , critical value of t is

tc =2.10 (from t table)

Thus for 95% confidence , upper limit is

=5.35+1.12

= 6.47

lower limit is

=5.35- 1.12

= 4.23

margin of error = = 1.12

(b) For using t distribution for construction of confidence interval we need : normal distribution of uric acid .

(c) Interpretation is

d- The probability that the interval contains true average uric acid level for this patient is 0.95

(e) Given E= 1.16

For 95% confidence , zc =1.96 (from z table)

Required sample size = 10

Note : Here we used z distribution as we donot know the sample size so we cannot find out degrees of freedom