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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