According to Advertising Age, the average base salary for women working as copywriters is $66,000. Assume that the salaries are normally distributed and that the standard deviation is $7,500.
a. What is the probability of a woman receiving a salary between $60,000 and $75,000?
b. What is the probability of a woman receiving a salary of less than $60,000?
c. What is the probability of a woman receiving salary of more than 80,000?
d. What is the probability of a woman receiving a salary between $75,000 and $80,000?
e. How much (or more) would a woman have to make to have the salary in the top (upper) 3%?
just use your Z table.
(a)
60000 is -.8σ
75000 is +1.2σ
P(Z < -.8) = .2119
P(Z < 1.2) = .8849
So, P(-.8 < Z < 1.2) = .6730
Do the others in like wise.
To solve these questions, we can use the Z-score formula and the standard normal distribution table.
a. To find the probability of a woman receiving a salary between $60,000 and $75,000, we need to calculate the Z-scores for both salaries and find the area under the curve between these two Z-scores.
Z1 = (60,000 - 66,000) / 7,500 = -0.8
Z2 = (75,000 - 66,000) / 7,500 = 1.2
Using the standard normal distribution table or calculator, we can find the area to the left of Z1 as 0.2119 and the area to the left of Z2 as 0.8849.
Therefore, the probability of a woman receiving a salary between $60,000 and $75,000 is the difference between the two probabilities:
Probability = 0.8849 - 0.2119 = 0.6730
b. To find the probability of a woman receiving a salary of less than $60,000, we calculate the Z-score for $60,000 and find the area to the left of this Z-score.
Z = (60,000 - 66,000) / 7,500 = -0.8
Using the standard normal distribution table or calculator, we can find the area to the left of Z as 0.2119.
Therefore, the probability of a woman receiving a salary less than $60,000 is 0.2119.
c. To find the probability of a woman receiving a salary of more than $80,000, we calculate the Z-score for $80,000 and find the area to the right of this Z-score.
Z = (80,000 - 66,000) / 7,500 = 1.8667
Using the standard normal distribution table or calculator, we can find the area to the right of Z as 0.0322.
Therefore, the probability of a woman receiving a salary more than $80,000 is 0.0322.
d. To find the probability of a woman receiving a salary between $75,000 and $80,000, we calculate the Z-scores for both salaries and find the area under the curve between these two Z-scores.
Z1 = (75,000 - 66,000) / 7,500 = 1.2
Z2 = (80,000 - 66,000) / 7,500 = 1.8667
Using the standard normal distribution table or calculator, we can find the area to the left of Z1 as 0.8849 and the area to the left of Z2 as 0.9671.
Therefore, the probability of a woman receiving a salary between $75,000 and $80,000 is the difference between the two probabilities:
Probability = 0.9671 - 0.8849 = 0.0822
e. To find the salary a woman would have to make to be in the top (upper) 3%, we need to find the Z-score corresponding to the top 3% area.
The top 3% area corresponds to the area to the left of Z in the standard normal distribution table. We can find this Z-score using the inverse normal distribution function or the Z-table.
Z = InvNorm(1 - 0.03) ≈ InvNorm(0.97) ≈ 1.8808
We can then calculate the salary using the Z-score formula:
X = Z * Standard Deviation + Mean
X = 1.8808 * 7,500 + 66,000 ≈ $79,356.50
Therefore, a woman would have to make at least $79,356.50 to have a salary in the top 3%.
To solve these probability questions, we need to use the properties of the normal distribution. Specifically, we will use the z-score formula and the standard normal distribution table.
a. To find the probability of a woman receiving a salary between $60,000 and $75,000, we need to calculate the z-scores for both salaries and then find the area under the standard normal curve between those z-scores.
First, calculate the z-score for $60,000:
z = (60,000 - 66,000) / 7,500
z = -0.8
Next, calculate the z-score for $75,000:
z = (75,000 - 66,000) / 7,500
z = 1.2
Using the standard normal distribution table or a software tool, find the probabilities corresponding to these z-scores. The probability of a woman receiving a salary between $60,000 and $75,000 is the difference between these two probabilities.
b. To find the probability of a woman receiving a salary less than $60,000, we calculate the z-score for $60,000 and find the corresponding probability from the standard normal distribution table.
c. To find the probability of a woman receiving a salary more than $80,000, we calculate the z-score for $80,000 and find the probability of the area to the right of this z-score from the standard normal distribution table.
d. To find the probability of a woman receiving a salary between $75,000 and $80,000, we calculate the z-scores for these two salaries and find the difference between their probabilities.
e. To find the salary necessary to be in the top 3%, we need to find the z-score corresponding to the area to the left of this percentile. Then use the z-score formula to solve for salary.
Remember to use the mean and standard deviation provided (mean = $66,000, standard deviation = $7,500) throughout the calculations.