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Homework 6 for Stat 3021 Section 004 1
STAT 3021 Introduction to Probability &
Statistics
Homework 6
Due Date: April 9
- An instructor knows from past experience that student examination scores have mean 70
and standard deviation 6. At present, the instructor is teaching two separate classes –
one of size of 36 and the other of size 64.
(a) Approximate the probability that the average test scores in the class of size of 36
lies between 60 and 80.
(b) Repeat (a) for the class of size 64.
(c) What is the approximate probability that the average test score in the class of size
36 is higher than that in the class of size 64.
Solution:
(a)
P (60 <
X
36
< 80) = P (
X
36
= P (
X
36
= P (− 10 < Z < 10) = 1.
(b)
P (60 <
X 64 < 80) = P (
X
64
= P (
X
64
= P (− 13. 3 < Z < 13 .3) = 1.
(c)
P (
X
36
X
64
) = P (
X
36
X
64
= P (
X
36
X
64
6
2
36
6
2
64
6
2
36
6
2
64
= P (Z > 0) = 0. 5.
2 Kuo-Jung Lee
- The mean score for freshmen on an aptitude test at a certain college is 540, with a
standard deviation of 50. What is the probability that two groups of students selected at
random, consisting of 32 and 50 students, respectively, will differ in their mean scores by
(a) more than 20 points?
(b) an amount between 5 and 10 points?
Solution:
(a)
P (|
X
32
X
50
| > 20) = P (
X
32
X
50
50
2
32
50
2
50
50
2
32
50
2
50
= P (|Z| > 1 .767) = 0. 077.
(b)
P (5 < |
X
32
X
50
= P
50
2
32
50
2
50
X
32
X
50
50
2
32
50
2
50
50
2
32
50
2
50
= P (0. 44 < |Z| < 0 .88) = 0. 282.
- Find the probability that a random sample of 25 observations, from a normal population
with variance σ
2
= 6, will have a variance s
2
(a) greater than 9.1;
(b) between 3.462 and 10.745.
Solution:
(a)
P (S
2
9 .1) = P
(n − 1) · S
2
σ
2
= P
χ
2
- 4
where χ
2
∼ χ
2
(ν = 24)
(b)
P (3. 462 < S
2
< 10 .745) = P
(n − 1) · S
2
σ
2
= P
- 848 < χ
2
< 42. 98
where χ
2
∼ χ
2
(ν = 24)
- Suppose scores on an IQ test are normally distributed, with a mean of 100. Suppose 100
people are randomly selected and tested. The standard deviation in the sample group is
- What is the probability that the average test score in the sample group will be at
