the average of USA university vice chancellors are normally distributed with mean 57 and variance 225.if a vice chancellor is chosen at randon what is the probability that his/her age is
(a):more than 52years
[b):less than 48years
(c):between 40 and 64 years
#so much
To find the probability in each case, we will use the concept of the standard normal distribution. We know that the average of the university vice chancellors is normally distributed with a mean of 57 and a variance of 225.
To solve these questions, we will convert the given values into z-scores using the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we are interested in
- μ is the mean
- σ is the standard deviation (which can be calculated by taking the square root of the variance)
(a): More than 52 years
To calculate the probability of the vice chancellor's age being more than 52 years, we need to find the area under the curve to the right of 52. We will calculate the z-score for 52 and then find the probability using a z-table.
z = (52 - 57) / sqrt(225)
z = -5 / 15
z = -1/3
Using a z-table or calculator, we find that the probability corresponding to a z-score of -1/3 is approximately 0.368.
(b): Less than 48 years
To calculate the probability of the vice chancellor's age being less than 48 years, we need to find the area under the curve to the left of 48. We will calculate the z-score for 48 and then find the probability using a z-table.
z = (48 - 57) / sqrt(225)
z = -9 / 15
z = -3/5
Using a z-table or calculator, we find that the probability corresponding to a z-score of -3/5 is approximately 0.274.
(c): Between 40 and 64 years
To calculate the probability of the vice chancellor's age being between 40 and 64 years, we need to find the area under the curve between these two values. We will calculate the z-scores for both 40 and 64, and then find the difference between their probabilities using a z-table.
For 40:
z1 = (40 - 57) / sqrt(225)
z1 = -17 / 15
z1 = -17/15
For 64:
z2 = (64 - 57) / sqrt(225)
z2 = 7 / 15
z2 = 7/15
Using a z-table or calculator, we find that the probability corresponding to a z-score of -17/15 is approximately 0.059 and the probability corresponding to a z-score of 7/15 is approximately 0.788.
Therefore, the probability of the vice chancellor's age being between 40 and 64 years is 0.788 - 0.059 = 0.729.
In conclusion,
(a) The probability that the vice chancellor's age is more than 52 years is approximately 0.368.
(b) The probability that the vice chancellor's age is less than 48 years is approximately 0.274.
(c) The probability that the vice chancellor's age is between 40 and 64 years is approximately 0.729.