statistics exam 2
An exam consists of 20 multiple-choice questions. Each question has five possible answers and all questions carry equal weight.
Mary has not studied for the exam and will be picking her answers at random.
What is the probability that Mary will get a score of at least 30% on this exam? n=20, p=0.20
P(X≥6)
(0.30)(20)=6
Distribution Plot
Binomial, n=20, p=0.2
0.25
Probability
0.20
0.15
0.10
0.05
0.1958
0.00
0
P(X≥6)=0.196
X
6
If 40% is the lowest passing score, what is the probability that Mary will pass the exam? n=20, p=0.20
(0.40)(20)=8
P(X≥8)
Distribution Plot
Binomial, n=20, p=0.2
0.25
Probability
0.20
0.15
0.10
0.05
0.00
0.03214
0
P(X≥8)=0.032
X
8
What is the probability that Mary will get a score of 25% on this exam? n=20, p=0.20
(0.25)(20)=5
P(X=5)
Distribution Plot
Binomial, n=20, p=0.2
0.25
0.20
Probability
0.1746
0.15
0.10
0.05
0.00
0
P(X=5)=0.175
5
X
10
What is the probability that Mary will get a score of no more than 35% on this exam? n=20, p=0.20
(0.35)(20)=7
P(X≤7)
Distribution Plot
Binomial, n=20, p=0.2
0.25
0.9679
Probability
0.20
0.15
0.10
0.05
0.00
X
P(X≤7)=0.968
7
10
The following information is available on the number of calls received at the telephone switchboard of the ABC Co. The number of calls received in any time interval is independent of the number of calls received in any other non-overlapping interval.
Additionally, it is known that the expected
(average) number of calls in an interval is proportional to the length of the interval.
Furthermore, it is known that the average number of calls received per hour is 180.
What is the most likely number of calls to be received at the telephone switchboard over a period of 1 minute?
180
(1) = 3
60
μ=
As shown on the next page, the highest probability is 0.2240 (using 4-decimal accuracy), and such probability is observed for x=2 and x=3. Thus,
P(X=2)=P(X=3)=0.2240
Probability Density Function
Poisson with mean = 3 x 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
P( X = x )
0.049787
0.149361
0.224042
0.224042
0.168031
0.100819
0.050409
0.021604
0.008102
0.002701
0.000810
0.000221
0.000055
0.000013
0.000003
0.000001
What is the probability that no more than 8 calls will be received at the telephone switchboard over a period of 2 minutes?
180
(2) =
60
μ=
6
Distribution Plot
Poisson, Mean=6
0.18
0.16
0.14
0.8472
Probability
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
P(X≤8)=0.8472
8
15
What is the probability that at least 1 call will be received at the telephone switchboard over a period of 30 seconds?
180 30
=1.5
60 60
μ=
Distribution Plot
Poisson, Mean=1.5
0.35
0.30
Probability
0.25
0.20
0.15
0.10
0.7769
0.05
0.00
0
1
P(X≥1)=0.7769
X
An insurance policy, issued by the MNP
Insurance Company, gives policyholders a payment on their 65th birthday and additional payments every three years thereafter
(provided that the policyholder is still alive).
The lifetimes of the policyholders are normally distributed with mean 73 years and standard deviation 5 years. age 65 68 71 74 77 80 83 86 89 92 •••
payment 1 2
3
4
5
6
7
8
9 10 •••
What proportion of the policyholders will receive exactly 4 payments?
P(74
Mary has not studied for the exam and will be picking her answers at random.
What is the probability that Mary will get a score of at least 30% on this exam? n=20, p=0.20
P(X≥6)
(0.30)(20)=6
Distribution Plot
Binomial, n=20, p=0.2
0.25
Probability
0.20
0.15
0.10
0.05
0.1958
0.00
0
P(X≥6)=0.196
X
6
If 40% is the lowest passing score, what is the probability that Mary will pass the exam? n=20, p=0.20
(0.40)(20)=8
P(X≥8)
Distribution Plot
Binomial, n=20, p=0.2
0.25
Probability
0.20
0.15
0.10
0.05
0.00
0.03214
0
P(X≥8)=0.032
X
8
What is the probability that Mary will get a score of 25% on this exam? n=20, p=0.20
(0.25)(20)=5
P(X=5)
Distribution Plot
Binomial, n=20, p=0.2
0.25
0.20
Probability
0.1746
0.15
0.10
0.05
0.00
0
P(X=5)=0.175
5
X
10
What is the probability that Mary will get a score of no more than 35% on this exam? n=20, p=0.20
(0.35)(20)=7
P(X≤7)
Distribution Plot
Binomial, n=20, p=0.2
0.25
0.9679
Probability
0.20
0.15
0.10
0.05
0.00
X
P(X≤7)=0.968
7
10
The following information is available on the number of calls received at the telephone switchboard of the ABC Co. The number of calls received in any time interval is independent of the number of calls received in any other non-overlapping interval.
Additionally, it is known that the expected
(average) number of calls in an interval is proportional to the length of the interval.
Furthermore, it is known that the average number of calls received per hour is 180.
What is the most likely number of calls to be received at the telephone switchboard over a period of 1 minute?
180
(1) = 3
60
μ=
As shown on the next page, the highest probability is 0.2240 (using 4-decimal accuracy), and such probability is observed for x=2 and x=3. Thus,
P(X=2)=P(X=3)=0.2240
Probability Density Function
Poisson with mean = 3 x 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
P( X = x )
0.049787
0.149361
0.224042
0.224042
0.168031
0.100819
0.050409
0.021604
0.008102
0.002701
0.000810
0.000221
0.000055
0.000013
0.000003
0.000001
What is the probability that no more than 8 calls will be received at the telephone switchboard over a period of 2 minutes?
180
(2) =
60
μ=
6
Distribution Plot
Poisson, Mean=6
0.18
0.16
0.14
0.8472
Probability
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
P(X≤8)=0.8472
8
15
What is the probability that at least 1 call will be received at the telephone switchboard over a period of 30 seconds?
180 30
=1.5
60 60
μ=
Distribution Plot
Poisson, Mean=1.5
0.35
0.30
Probability
0.25
0.20
0.15
0.10
0.7769
0.05
0.00
0
1
P(X≥1)=0.7769
X
An insurance policy, issued by the MNP
Insurance Company, gives policyholders a payment on their 65th birthday and additional payments every three years thereafter
(provided that the policyholder is still alive).
The lifetimes of the policyholders are normally distributed with mean 73 years and standard deviation 5 years. age 65 68 71 74 77 80 83 86 89 92 •••
payment 1 2
3
4
5
6
7
8
9 10 •••
What proportion of the policyholders will receive exactly 4 payments?
P(74