Answer:
a) 0.434
b) 0.983
c) 0.367
Step-by-step explanation:
The exact question with the given parameters wasn't obtained online, but the same question, albeit with different parameters is then obtained. Hopefully, this Helps to solve the complete question with the required parameters.
Antique Accents tracks their daily profits and has found that the distribution of profis is approximately normal with a mean of $17,700.00 and a standard deviation of about $900.00. Using this information, answer the following questions For full marks your answer should be accurate to at least three decimal places. Compute the probability that tomorrow's profit will be
a) less than $16,791 or greater than $18,231
b) greater than $15,783
c) between $17,997 and $20,130
Solution
This is a normal distribution problem with
Mean = μ = $17,700
Standard deviation = σ = $900
a) less than $16,791 or greater than $18,231. P(x < 16,791) or P(X > 18,231) = P(X < 16,791) + P(x > 18,231)
We first standardize 16,791 and 18,231
The standardized score for any value is the value minus the mean then divided by the standard deviation.
For 16791
z = (x - μ)/σ = (16791 - 17700)/900 = - 1.01
For 18231
z = (x - μ)/σ = (18231 - 17700)/900 = 0.59
To determine the required probability
P(X < 16,791) + P(x > 18,231) = P(z < -1.01) + P(z > 0.59)
We'll use data from the normal probability table for these probabilities
P(X < 16,791) + P(x > 18,231) = P(z < -1.01) + P(z > 0.59)
P(z < -1.01) = 0.15625
P(z > 0.59) = 1 - (z ≤ 0.59) = 1 - 0.7224 = 0.2776
P(X < 16,791) + P(x > 18,231) = P(z < -1.01) + P(z > 0.59) = 0.15625 + 0.2776 = 0.43385 = 0.434 to 3 d.p
b) greater than $15,783. P(x > 15783)
We standardize 15783
z = (x - μ)/σ = (15783 - 17700)/900 = -2.13
To determine the required probability
P(x > 15783) = P(z > -2.13)
We'll use data from the normal probability table for this probability
P(x > 15783) = P(z > -2.13) = 1 - P(z ≤ - 2.13)
= 1 - 0.01659 = 0.98341 = 0.983 to 3 d.p.
c) between $17,997 and $20,130.
P(17,997 < x < 20,130)
We first standardize 17,997 and 20,130
The standardized score for any value is the value minus the mean then divided by the standard deviation.
For 17,997
z = (x - μ)/σ = (17,997 - 17700)/900 = 0.33
For 20,130
z = (x - μ)/σ = (20,130 - 17700)/900 = 2.70
To determine the required probability
P(17,997 < x < 20,130) = P(0.33 < x < 2.70)
We'll use data from the normal probability table for these probabilities
P(17,997 < x < 20,130) = P(0.33 < x < 2.70)
= P(z < 2.70) - P(z < 0.33)
= 0.99653 - 0.62930
= 0.36723 = 0.367 to 3 d.p.
Hope this Helps!!!