2. In a given year the Murphy College admissions office accepts applications from students, who are then either accepted or not-accepted. Accepted students may or may not decide to attend Murphy College. Suppose the scores on a certain entrance exam for applicants of Murphy College follow the normal distribution with mean 1600 and standard deviation 300.
a. What is the probability that a given applicant has a score of more than 2050?
b. What is the probability that a given applicant has a score of between 1450 and 1900?
c. Suppose that top scoring applicants are award merit scholarship money based on their scores. Suppose that any student scoring above 1900 is awarded $20,000 in scholarship money. If 1000 students apply to Murphy College in a given year, how much money does the college award in total?
d. How would Murphy College calculate the expected amount (in $) of merit scholarships dispersed in a given year?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To answer these questions, we need to use the concept of z-scores and the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. By converting our raw scores to z-scores, we can find the corresponding probabilities using a normal distribution table or a calculator.
a. To find the probability that a given applicant has a score of more than 2050, we need to find the area to the right of 2050 on the standard normal distribution curve. First, we need to calculate the z-score for 2050 using the formula:
z = (x - mean) / standard deviation
In this case, x (the score) is 2050, the mean is 1600, and the standard deviation is 300. Plugging in the values, we get:
z = (2050 - 1600) / 300 = 1.5
Using a standard normal distribution table or calculator, the probability corresponding to a z-score of 1.5 is approximately 0.9332. Therefore, the probability that a given applicant has a score of more than 2050 is 0.9332 (or 93.32%).
b. To find the probability that a given applicant has a score between 1450 and 1900, we need to find the area between these two scores on the standard normal distribution curve. First, we will calculate the z-scores for 1450 and 1900 using the formula mentioned before:
For 1450:
z = (1450 - 1600) / 300 = -0.5
For 1900:
z = (1900 - 1600) / 300 = 1
Using a standard normal distribution table or calculator, we can find the probabilities corresponding to these z-scores. The probability for a z-score of -0.5 is approximately 0.3085, and the probability for a z-score of 1 is approximately 0.8413. To find the probability between these two scores, we subtract the probability for the lower z-score from the probability for the higher z-score:
0.8413 - 0.3085 = 0.5328
Therefore, the probability that a given applicant has a score between 1450 and 1900 is 0.5328 (or 53.28%).
c. If any student scoring above 1900 is awarded $20,000 in scholarship money, we can find the number of students who qualify for the scholarship by finding the area to the right of 1900 on the standard normal distribution curve.
Using the z-score formula, we find the z-score for 1900:
z = (1900 - 1600) / 300 = 1
Using a standard normal distribution table or calculator, the probability corresponding to a z-score of 1 is approximately 0.8413. Therefore, the probability that a given applicant has a score greater than 1900 (and qualifies for the scholarship) is 0.8413.
If there are 1000 students applying, the number of students who qualify for the scholarship is:
1000 * 0.8413 = 841.3
Since we can't have a fraction of a student, we round down to 841. Therefore, Murphy College would award a total of $20,000 in scholarship money to 841 students.
d. To calculate the expected amount (in $) of merit scholarships dispersed in a given year, Murphy College would multiply the probability of qualifying for the scholarship (found in part c, which is 0.8413) by the value of the scholarship ($20,000):
Expected amount = Probability * Value = 0.8413 * $20,000 = $16,826
Therefore, the expected amount of merit scholarships dispersed in a given year at Murphy College is $16,826.