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The incomes in a certain large population of college teachers have a normal distribution with mean $75,000 and standard deviation $10,000. Sixteen (16) teachers are selected at random from this population to serve on a committee. What is the probability that their average salary is more than $77,500?

A. 0.0228
B. 0.1587
C. 0.8413
D. Essentially 0

Respuesta :

Answer:

Option B) 0.1587

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = $75,000

Standard Deviation, σ = $10,000

We are given that the distribution of salary of college teachers is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

Standard error due to sampling:

[tex]=\dfrac{\sigma}{\sqty{n}} = \dfrac{10000}{\sqrt{16}} = 2500[/tex]

P(average salary is more than $77,500)

P(x > 77500)

[tex]P( x > 77500) = P( z > \displaystyle\frac{77500 - 75000}{2500}) = P(z > 1)[/tex]

[tex]= 1 - P(z \leq1)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(x > 77500) = 1 - 0.8413 = 0.1587 = 15.87\%[/tex]

Option B) 0.1587

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