Quiz 7 - Duke University
STA 110B
Name
Spring 2000
Section
Quiz 7
week of 20MAR2000
1. The administrators for a hospital wished to estimate the average number of days required for
treatment of patients between the ages of 25 and 34. A random sample of 500 hospital patients
between these ages produced a sample mean and sample standard deviation equal to 5.4 and 3.1
days respectively.
a. (3 points) Construct a 95% confidence interval for the mean length of stay for the population of
patients from which the sample was drawn.
Since the t distribution with n ? 1 = 500 ? 1 = 499 degrees of freedom is approximately equal
to the normal distribution, we can use z.025 here.
s
x? ¡À z.025 ¡Ì
n
3.1
5.4 ¡À 1.96 ¡Ì
500
(5.1 , 5.7)
b. (3 points) If we wanted the error of estimation to be less than or equal to 0.15, what would be
the minimum sample size needed?
s
z.025 ¡Ì
n
3.1
1.96 ¡Ì
n
1
¡Ì
n
= 0.15
= 0.15
=
0.15
(1.96)(3.1)
(1.96)(3.1)
0.15
n ¡Ö 1640.8
n =
2
So, the minimum sample size needed is 1641, which will actually give you an error of estimation
slightly less than 0.15.
2. (3 points) The scores of a random sample of 9 people who took the Graduate Record Examination
had a sample mean of 540 and a sample stadard deviation of 50.
a. (3 points) Construct a 99% confidence interval for the population mean GRE score.
s
x? ¡À tn?1
.005 ¡Ì
n
50
540 ¡À 3.36 ¡Ì
9
(484 , 596)
b. (1 point) What assumption(s) is/are necessary for this method to be valid?
The small sample confidence interval using the t distribution is based on the assumption that
GRE test scores are normally distributed. However, the method gives a very good approximation
(is ¡°robust¡±) even when the sample size is small as long as the distribution is fairly mound-shaped
(no major departures from normality are allowed when the sample size is small). Therfore, I will
also accept answers along this line of reasoning.
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