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 nine 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 = 1.81 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 n is large normal distribution of uric acid σ is unknown

(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 0.95. We are 95% confident that the true uric acid level for this patient falls within this interval.     The probability that this interval contains the true average uric acid level for this patient is 0.05. We are 5% confident that the true uric acid level for this patient falls within this interval.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.06 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)

sample mean, xbar = 5.35
sample standard deviation, σ = 1.81
sample size, n = 9

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 1.81/sqrt(9)
ME = 1.18

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (5.35 - 1.96 * 1.81/sqrt(9) , 5.35 + 1.96 * 1.81/sqrt(9))
CI = (4.17 , 6.53)

lower limit = 4.17    
upper limit     = 6.53

margin of error = 1.18

b)
σ is known n is large normal distribution

c)

We are 95% confident that the true uric acid level for this patient falls within this interval.

d)
The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 1.06, σ = 1.81

The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 1.81/1.06)^2
n = 11.2

Therefore, the sample size needed to satisfy the condition n >= 11.2 and it must be an integer number, we conclude that the minimum required sample size is n = 12
Ans : Sample size, n = 12 or 11