Respuesta :
Answer:
(a) P(X > $57,000) = 0.0643
(b) P(X < $46,000) = 0.1423
(c) P(X > $40,000) = 0.0066
(d) P($45,000 < X < $54,000) = 0.6959
Step-by-step explanation:
We are given that U.S. Bureau of Economic Statistics reports that the average annual salary in the metropolitan Boston area is $50,542.
Suppose annual salaries in the metropolitan Boston area are normally distributed with a standard deviation of $4,246.
Let X = annual salaries in the metropolitan Boston area
SO, X ~ Normal([tex]\mu=$50,542,\sigma^{2} = $4,246^{2}[/tex])
The z-score probability distribution for normal distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = average annual salary in the Boston area = $50,542
[tex]\sigma[/tex] = standard deviation = $4,246
(a) Probability that the worker’s annual salary is more than $57,000 is given by = P(X > $57,000)
P(X > $57,000) = P( [tex]\frac{X-\mu}{\sigma }[/tex] > [tex]\frac{57,000-50,542}{4,246 }[/tex] ) = P(Z > 1.52) = 1 - P(Z [tex]\leq[/tex] 1.52)
= 1 - 0.93574 = 0.0643
The above probability is calculated by looking at the value of x = 1.52 in the z table which gave an area of 0.93574.
(b) Probability that the worker’s annual salary is less than $46,000 is given by = P(X < $46,000)
P(X < $46,000) = P( [tex]\frac{X-\mu}{\sigma }[/tex] < [tex]\frac{46,000-50,542}{4,246 }[/tex] ) = P(Z < -1.07) = 1 - P(Z [tex]\leq[/tex] 1.07)
= 1 - 0.85769 = 0.1423
The above probability is calculated by looking at the value of x = 1.07 in the z table which gave an area of 0.85769.
(c) Probability that the worker’s annual salary is more than $40,000 is given by = P(X > $40,000)
P(X > $40,000) = P( [tex]\frac{X-\mu}{\sigma }[/tex] > [tex]\frac{40,000-50,542}{4,246 }[/tex] ) = P(Z > -2.48) = P(Z < 2.48)
= 1 - 0.99343 = 0.0066
The above probability is calculated by looking at the value of x = 2.48 in the z table which gave an area of 0.99343.
(d) Probability that the worker’s annual salary is between $45,000 and $54,000 is given by = P($45,000 < X < $54,000)
P($45,000 < X < $54,000) = P(X < $54,000) - P(X [tex]\leq[/tex] $45,000)
P(X < $54,000) = P( [tex]\frac{X-\mu}{\sigma }[/tex] < [tex]\frac{54,000-50,542}{4,246 }[/tex] ) = P(Z < 0.81) = 0.79103
P(X [tex]\leq[/tex] $45,000) = P( [tex]\frac{X-\mu}{\sigma }[/tex] [tex]\leq[/tex] [tex]\frac{45,000-50,542}{4,246 }[/tex] ) = P(Z [tex]\leq[/tex] -1.31) = 1 - P(Z < 1.31)
= 1 - 0.90490 = 0.0951
The above probability is calculated by looking at the value of x = 0.81 and x = 1.31 in the z table which gave an area of 0.79103 and 0.9049 respectively.
Therefore, P($45,000 < X < $54,000) = 0.79103 - 0.0951 = 0.6959
