Question

20 patients tested for the Urea level in their blood. They have an average Urea value of 65 mg/dL with a standard deviation of 8.5 mg/dL. Normal urea level is 40 mg/dL.
Is the value of urea is significantly more than the normal expected value of Urea at 90% confidence?

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 40
Alternative Hypothesis, Ha: μ > 40

Rejection Region
This is right tailed test, for α = 0.1 and df = 19
Critical value of t is 1.328.
Hence reject H0 if t > 1.328

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (65 - 40)/(8.5/sqrt(20))
t = 13.153

P-value Approach
P-value = 0
As P-value < 0.1, reject the null hypothesis.

yes, the value of urea is significantly more than the normal expected value of Urea at 90% confidence

sample mean, xbar = 65
sample standard deviation, s = 8.5
sample size, n = 20
degrees of freedom, df = n - 1 = 19

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, tc = t(α/2, df) = 1.729

ME = tc * s/sqrt(n)
ME = 1.729 * 8.5/sqrt(20)
ME = 3.286

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (65 - 1.729 * 8.5/sqrt(20) , 65 + 1.729 * 8.5/sqrt(20))
CI = (61.71 , 68.29)
AS the confidence interval does not contains 40 so,  the value of urea is significantly more than the normal expected value of Urea at 90% confidence