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 five 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 = 1.91 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, n is large, σ is unknown, normal distribution of uric acid, σ is known

(c) Interpret your results in the context of this problem. The probability that this interval contains the true average uric acid level for this patient is 0.05. There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient. There is not enough information to make an interpretation. There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient. The probability that this interval contains the true average uric acid level for this patient is 0.95.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.20 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

Solution :

a) The 95% confidence interval for population mean is given as follows :

Where, x̅ is sample mean, σ is population standard deviation, n is sample size and Z(0.05/2) is critical z-value to construct 95% confidence interval.

We have,  x̅ = 5.35 mg/dl, σ = 1.91 mg/dl, n = 5

Using Z-table we get, Z(0.05/2) = 1.96

Hence, 95% confidence interval for the population mean concentration of uric acid in this patient's blood is,

Lower limit = 3.68

Upper limit = 7.02

Margin of error = 1.67

b) Assumption :

1) Normal distribution of uric acid,

2) σ is known

c) Interpretation :

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) The sample size is given as follows :

Where, E is margin of error.

We have, E = 1.20,  σ = 1.91 and Z(0.05/2) = 1.96

Hence, required sample size is,

On rounding to nearest whole number we get, n = 10

Hence, the necessary sample size is 10.