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bekas [8.4K]
3 years ago
10

According to Advertising Age, the average base salary for women working as copywriters in advertising firms is higher than the a

verage base salary for men. The average base salary for women is $67,000 and the average base salary for men is $65,500 (Working Woman, July/August 2000). Assume salaries are normally distributed and that the standard deviation is $7000 for both men and women.
Required:
a. What is the probability of a woman receiving a salary in excess of $75,000 (to 4 decimals)?
b. What is the probability of a man receiving a salary in excess of $75,000 (to 4 decimals)?
c. What is the probability of a woman receiving a salary below $50,000 (to 4 decimals)?
d. How much would a woman have to make to have a higher salary than 99% of her male counterparts (0 decimals)?
Mathematics
1 answer:
horrorfan [7]3 years ago
7 0

Answer:

(a) The probability of a woman receiving a salary in excess of $75,000 is 0.1271.

(b) The probability of a man receiving a salary in excess of $75,000 is 0.0870.

(c) The probability of a woman receiving a salary below $50,000 is 0.9925.

(d) A woman would have to make a higher salary of $81,810 than 99% of her male counterparts.

Step-by-step explanation:

Let the random variable <em>X</em> represent the salary for women and <em>Y</em> represent the salary for men.

It is provided that:

X\sim N(67000, 7000^{2})\\\\Y\sim N(65500, 7000^{2})

(a)

Compute the probability of a woman receiving a salary in excess of $75,000 as follows:

P(X>75000)=P(\frac{X-\mu_{x}}{\sigma_{x}}>\frac{75000-67000}{7000})

                     =P(Z>1.14)\\\\=1-P(Z

Thus, the probability of a woman receiving a salary in excess of $75,000 is 0.1271.

(b)

Compute the probability of a man receiving a salary in excess of $75,000 as follows:

P(Y>75000)=P(\frac{Y-\mu_{y}}{\sigma_{y}}>\frac{75000-65500}{7000})

                     =P(Z>1.36)\\\\=1-P(Z

Thus, the probability of a man receiving a salary in excess of $75,000 is 0.0870.

(c)

Compute the probability of a woman receiving a salary below $50,000 as follows:

P(X

                     =P(Z>-2.43)\\\\=P(Z

Thus, the probability of a woman receiving a salary below $50,000 is 0.9925.

(d)

Let <em>a</em> represent the salary a woman have to make to have a higher salary than 99% of her male counterparts.

Then,

   P(Y\leq a)=0.99

\Rightarrow P(Z

The <em>z</em>-score for this probability is:

<em>z</em>-score = 2.33

Compute the value of <em>a</em> as follows:

\frac{a-\mu_{y}}{\sigma_{y}}=2.33\\\\

    a=\mu_{y}+(2.33\times \sigma_{y})\\\\

       =65500+(2.33\times7000)\\\\=65500+16310\\\\=81810

Thus, a woman would have to make a higher salary of $81,810 than 99% of her male counterparts.

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