A recent national survey of music fans aged 14 to 24 reveale

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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Solution

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