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 - 2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people.What is the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages?A. 0.14 B. 0.38 C. 0.57 D. 0.77 E. 0.88

Answer 1

Answer:

The correct answer is D. 0.77

Step-by-step explanation:

$X_i \text{ is the difference between true and reported age. where i = 1,2,3......48 }\\\text{The rounded age are uniformly distributed. So, }X_i = 0\\\\and,\thinspace O_{X_i}^2=\frac{\text{(Upper Bound - Lower Bound})^2}{5\times 2.5\text{ ( Rounded value )}}\\\\O_{X_i}^2=\frac{(2.5 + 2.5)^2}{12}\approx 2.083\\\\O_X_i=1.443$

$\bar{X_{48}}=\frac{X_1+X_2+.......+X_{48}}{48}\\\\\bar{X_{48}}=\frac{\frac{48}{X_i}}{48}=0\\\\O_{\bar{X}_{48}}^2=\frac{48\cdot O_{X_i}^2}{48^2}=\frac{O_{X_i}^2}{48}\\\\\text{So, }\bar{X_i}\text{is approximately normal with mean 0 and standard deviation = }\\\\\frac{1.443}{\sqrt{48}}\approx 0.2083$

$=Pr[\frac{-1}{4}\leq \bar{X}_{48}\leq \frac{1}{4}]\\\\=Pr[\frac{-0.25}{0.2083}\leq Z\leq \frac{0.25}{0.2083}]\\\\=Pr[-1.2\leq Z\leq 1.2]\\=Pr(Z\leq 1.2)-Pr(Z\leq -1.2)\\=Pr(Z\leq 1.2)-(1-Pr(Z\leq 1.2))\\=2\times 0.8849 -1\text{ ( Z value for 1.2 is 0.8849 )}\\\approx 0.77$

Hence, the correct answer is 0.77