Questions: According to the National Association of Colleges and Employers, the 2015 average starting salary for new college graduates in health sciences was 51,541. The average starting salary for new college graduates in business was 53,901 (National Association of Colleges and Employers website). Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in health sciences is 11,000. Assume that the standard deviation for starting salaries for new college graduates in business is 15,000. a. What is the probability that a new college graduate in business will earn a starting salary of at least 65,000 (to 4 decimals)? b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least 65,000 (to 4 decimals)? c. What is the probability that a new college graduate in health sciences will earn a starting salary less than 40,000 (to 4 decimals)? d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences (to the nearest whole number)?

According to the National Association of Colleges and Employers, the 2015 average starting salary for new college graduates in health sciences was 51,541. The average starting salary for new college graduates in business was 53,901 (National Association of Colleges and Employers website). Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in health sciences is 11,000. Assume that the standard deviation for starting salaries for new college graduates in business is 15,000. a. What is the probability that a new college graduate in business will earn a starting salary of at least 65,000 (to 4 decimals)? b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least 65,000 (to 4 decimals)? c. What is the probability that a new college graduate in health sciences will earn a starting salary less than 40,000 (to 4 decimals)? d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences (to the nearest whole number)?
Transcript text: According to the National Association of Colleges and Employers, the 2015 average starting salary for new college graduates in health sciences was $51,541. The average starting salary for new college graduates in business was $53,901 (National Association of Colleges and Employers website). Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in health sciences is $11,000. Assume that the standard deviation for starting salaries for new college graduates in business is $15,000. a. What is the probability that a new college graduate in business will earn a starting salary of at least $65,000 (to 4 decimals)? b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least $65,000 (to 4 decimals)? c. What is the probability that a new college graduate in health sciences will earn a starting salary less than $40,000 (to 4 decimals)? d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences (to the nearest whole number)?
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Solution

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Solution Steps

Step 1: Probability for Business Graduates

To find the probability that a new college graduate in business will earn a starting salary of at least \$65,000, we calculate:

\[ P(X \geq 65000) = 1 - P(X < 65000) = 1 - \Phi(Z_{end}) = 1 - \Phi(0.7399) \]

Using the cumulative distribution function (CDF) of the standard normal distribution, we find:

\[ P(X \geq 65000) = 1 - 0.2297 = 0.7703 \]

Step 2: Probability for Health Sciences Graduates

Next, we calculate the probability that a new college graduate in health sciences will earn a starting salary of at least \$65,000:

\[ P(X \geq 65000) = 1 - P(X < 65000) = 1 - \Phi(Z_{end}) = 1 - \Phi(1.2235) \]

Again, using the CDF:

\[ P(X \geq 65000) = 1 - 0.1106 = 0.8894 \]

Step 3: Probability for Health Sciences Graduates Earning Less than \$40,000

Finally, we find the probability that a new college graduate in health sciences will earn a starting salary less than \$40,000:

\[ P(X < 40000) = \Phi(Z_{start}) - \Phi(-\infty) = \Phi(-1.0492) \]

Thus, we have:

\[ P(X < 40000) = 0.147 \]

Final Answer

  • Probability (Business >= \$65,000): \( \boxed{0.7703} \)
  • Probability (Health Sciences >= \$65,000): \( \boxed{0.8894} \)
  • Probability (Health Sciences < \$40,000): \( \boxed{0.147} \)
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