Question

In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from -2.5yrs to +2.5yrs. The healthcare data are based on a random sample of 48 people.What is the approximate probability that the mean of the rounded ages within 0.25 years of the mean of the true ages?

Answer 1

Answer:

The approximate probability that the mean of the rounded ages within 0.25 years of the mean of the true ages is P=0.766.

Step-by-step explanation:

We have a uniform distribution from which we are taking a sample of size n=48. We have to determine the sampling distribution and calculate the probability of getting a sample within 0.25 years of the mean of the true ages.

The mean of the uniform distribution is:

$\mu=\dfrac{Max+Min}{2}=\dfrac{2.5+(-2.5)}{2}=0$

The standard deviation of the uniform distribution is:

$\sigma=\dfrac{Max-Min}{\sqrt{12}}=\dfrac{2.5-(-2.5)}{\sqrt{12}}=\dfrac{5}{3.46}=1.44$

The sampling distribution can be approximated as a normal distribution with the following parameters:

$\mu_s=\mu=0\\\\\sigma_s=\dfrac{\sigma}{\sqrt{n}}=\dfrac{1.44}{\sqrt{48}}=\dfrac{1.44}{6.93}=0.21$

We can now calculate the probability that the sample mean falls within 0.25 from the mean of the true ages using the z-score:

$z=\dfrac{X-\mu}{\sigma}=\dfrac{0.25-0}{0.21}=\dfrac{0.25}{0.21}=1.19\\\\\\P(|X_s-\mu|$