Question

According to the Agency for Healthcare Research and Quality, the average length of a hospital stay is 4.5 days. Hospitals can reap many benefits by decreasing the length of patient stays in the hospital. A random sample of 7,200 stays in an Idaho hospital had an average stay of 4.2 days with a standard deviation of 8.3 days. Determine whether there is evidence that the Idaho hospital has an average stay that is lower than the national average. State your conclusion using a significance level of 0.01.

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 4.5
Alternative Hypothesis, Ha: μ < 4.5

Rejection Region
This is left tailed test, for α = 0.01 and df = 7199
Critical value of t is -2.327.
Hence reject H0 if t < -2.327

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (4.2 - 4.5)/(8.3/sqrt(7200))
t = -3.067

P-value Approach
P-value = 0.0011

As P-value < 0.01, reject the null hypothesis.

There is sufficient evidence to conclude that the he Idaho hospital has an average stay that is lower than the national average.