Respuesta :
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:
[tex]\mu=\dfrac{Max+Min}{2}=\dfrac{2.5+(-2.5)}{2}=0[/tex]
The standard deviation of the uniform distribution is:
[tex]\sigma=\dfrac{Max-Min}{\sqrt{12}}=\dfrac{2.5-(-2.5)}{\sqrt{12}}=\dfrac{5}{3.46}=1.44[/tex]
The sampling distribution can be approximated as a normal distribution with the following parameters:
[tex]\mu_s=\mu=0\\\\\sigma_s=\dfrac{\sigma}{\sqrt{n}}=\dfrac{1.44}{\sqrt{48}}=\dfrac{1.44}{6.93}=0.21[/tex]
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:
[tex]z=\dfrac{X-\mu}{\sigma}=\dfrac{0.25-0}{0.21}=\dfrac{0.25}{0.21}=1.19\\\\\\P(|X_s-\mu|<0.25)=P(|z|<1.19)=0.766[/tex]
