A recent national survey of music fans aged 14 to 24 revealed that about 36% of
ID: 3437401 • Letter: A
Question
A recent national survey of music fans aged 14 to 24 revealed that about 36% of them listen to streamed music on their computers every day. A sample of 20 music fans from the same age group is randomly selected. Show your work or calculator command to answer the following questions.
a. Explain what kind of probability distribution this situation satisfies. (Be sure to show all four requirements of this probability distribution.)
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b. What is the probability that at most 8 music fans from the sample listen to streamed music on their computers every day?
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c. What is the probability that at least 14 music fans from the sample listen to streamed music on their computers every day?
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d. What is the probability that exactly 6 music fans from the sample listen to streamed music on their computers every day?
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e. Find the mean (expected value) and the standard deviation of this distribution.
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Explanation / Answer
A recent national survey of music fans aged 14 to 24 revealed that about 36% of them listen to streamed music on their computers every day. A sample of 20 music fans from the same age group is randomly selected. Show your work or calculator command to answer the following questions.
1: The number of observations n is fixed.
2: Each observation is independent.
3: Each observation represents one of two outcomes ("success" or "failure").
4: The probability of "success" p is the same for each outcome.
n=20
p=0.36 ( only two events, listening or not listening)
The probability of listening music is same and n is fixed. Samples are independent and p is same for all cases.
Binomial distribution is used.
P( X 8) = 0.7317
P( X 14) = 0.0021
d. What is the probability that exactly 6 music fans from the sample listen to streamed music on their computers every day?
P( X =6) = 0.1632
e. Find the mean (expected value) and the standard deviation of this distribution.
Expectation = np = 7.2
Variance = np(1 - p) = 4.608
Standard deviation = 2.1466
Binomial Probabilities
Data
Sample size
20
Probability of an event of interest
0.36
Statistics
Mean
7.2
Variance
4.6080
Standard deviation
2.1466
Binomial Probabilities Table
X
P(X)
P(<=X)
P(<X)
P(>X)
P(>=X)
0
0.0001
0.0001
0.0000
0.9999
1.0000
1
0.0015
0.0016
0.0001
0.9984
0.9999
2
0.0080
0.0096
0.0016
0.9904
0.9984
3
0.0270
0.0366
0.0096
0.9634
0.9904
4
0.0645
0.1011
0.0366
0.8989
0.9634
5
0.1161
0.2171
0.1011
0.7829
0.8989
6
0.1632
0.3803
0.2171
0.6197
0.7829
7
0.1836
0.5639
0.3803
0.4361
0.6197
8
0.1678
0.7317
0.5639
0.2683
0.4361
9
0.1259
0.8576
0.7317
0.1424
0.2683
10
0.0779
0.9355
0.8576
0.0645
0.1424
11
0.0398
0.9753
0.9355
0.0247
0.0645
12
0.0168
0.9921
0.9753
0.0079
0.0247
13
0.0058
0.9979
0.9921
0.0021
0.0079
14
0.0016
0.9996
0.9979
0.0004
0.0021
15
0.0004
0.9999
0.9996
0.0001
0.0004
16
0.0001
1.0000
0.9999
0.0000
0.0001
17
0.0000
1.0000
1.0000
0.0000
0.0000
18
0.0000
1.0000
1.0000
0.0000
0.0000
19
0.0000
1.0000
1.0000
0.0000
0.0000
20
0.0000
1.0000
1.0000
0.0000
0.0000
Binomial Probabilities
Data
Sample size
20
Probability of an event of interest
0.36
Statistics
Mean
7.2
Variance
4.6080
Standard deviation
2.1466
Binomial Probabilities Table
X
P(X)
P(<=X)
P(<X)
P(>X)
P(>=X)
0
0.0001
0.0001
0.0000
0.9999
1.0000
1
0.0015
0.0016
0.0001
0.9984
0.9999
2
0.0080
0.0096
0.0016
0.9904
0.9984
3
0.0270
0.0366
0.0096
0.9634
0.9904
4
0.0645
0.1011
0.0366
0.8989
0.9634
5
0.1161
0.2171
0.1011
0.7829
0.8989
6
0.1632
0.3803
0.2171
0.6197
0.7829
7
0.1836
0.5639
0.3803
0.4361
0.6197
8
0.1678
0.7317
0.5639
0.2683
0.4361
9
0.1259
0.8576
0.7317
0.1424
0.2683
10
0.0779
0.9355
0.8576
0.0645
0.1424
11
0.0398
0.9753
0.9355
0.0247
0.0645
12
0.0168
0.9921
0.9753
0.0079
0.0247
13
0.0058
0.9979
0.9921
0.0021
0.0079
14
0.0016
0.9996
0.9979
0.0004
0.0021
15
0.0004
0.9999
0.9996
0.0001
0.0004
16
0.0001
1.0000
0.9999
0.0000
0.0001
17
0.0000
1.0000
1.0000
0.0000
0.0000
18
0.0000
1.0000
1.0000
0.0000
0.0000
19
0.0000
1.0000
1.0000
0.0000
0.0000
20
0.0000
1.0000
1.0000
0.0000
0.0000