thank you in advance!
in a given year the college admissions office accepts applicants from students, who are then either accepted or not accepted. accepted students may or may not decide to attend 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.
whats the probability that a given applicant has a score of more than 2050?
What is the probability that a given applicant has a score of between 1450 and 1900?
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?
How would 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 those Z scores.
To answer these questions, we can use the concept of the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. We can transform an individual's score using the formula:
Z = (X - μ) / σ
Where Z is the standard score (a measure of how many standard deviations a data point is from the mean), X is the individual's score, μ is the mean score, and σ is the standard deviation.
Now let's calculate the probabilities and expected amount of scholarship money:
1. Probability of a score greater than 2050:
To find the probability of a score greater than 2050, we need to find the area under the standard normal curve to the right of 2050. To do this, we need to standardize the score using the Z-score formula:
Z = (2050 - 1600) / 300
Z = 1.5
Using a standard normal distribution table or a calculator, we can find that the probability of getting a Z-score of 1.5 or greater is approximately 0.0668. This means that there is a 6.68% chance that a given applicant will score above 2050.
2. Probability of a score between 1450 and 1900:
Similarly, to find the probability of a score between 1450 and 1900, we need to find the area under the standard normal curve between these two scores. First, we calculate the Z-scores for both scores:
For 1450:
Z1 = (1450 - 1600) / 300
Z1 = -0.5
For 1900:
Z2 = (1900 - 1600) / 300
Z2 = 1
Using the standard normal distribution table or a calculator, we find the area to the left of Z1 is approximately 0.3085 and the area to the left of Z2 is approximately 0.8413. Therefore, the probability of obtaining a score between 1450 and 1900 is the difference between these two probabilities:
P(1450 < X < 1900) = P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1) = 0.8413 - 0.3085 = 0.5328
So, there is approximately a 53.28% chance that a given applicant will score between 1450 and 1900.
3. Total amount of scholarship money awarded:
Given that any student scoring above 1900 receives $20,000 in scholarship money, we need to calculate the number of students who score above 1900. For a normal distribution, the proportion of students scoring above 1900 can be found using the Z-score:
Z = (1900 - 1600) / 300
Z = 1
Using the standard normal distribution table or a calculator, we find that the area to the left of Z = 1 is approximately 0.8413. Therefore, approximately 84.13% of the applicants score below 1900, and the remaining 15.87% score above 1900.
Since we know that 1000 students apply to Murphy College in a given year, the number of students who score above 1900 would be:
Number of students above 1900 = 0.1587 * 1000 = 158.7
So, 159 students will receive $20,000 in scholarship money.
Therefore, the total amount of scholarship money awarded in a given year would be:
Total Scholarship Money = Number of students above 1900 * Amount awarded per student
Total Scholarship Money = 159 * $20,000 = $3,180,000
Hence, the college would award a total of $3,180,000 in scholarship money in that year.
4. Expected amount of scholarship money dispersed:
To calculate the expected amount of merit scholarships dispersed, we need to determine the probability of each scholarship amount being awarded and then multiply each amount by its probability. From our previous calculations, we know that the probability of receiving a $20,000 scholarship is 15.87% (0.1587) and the probability of not receiving any scholarship for those scoring below 1900 is 84.13% (0.8413).
Therefore, the expected amount of scholarship money dispersed can be calculated as follows:
Expected Scholarship Money = (Probability of $20,000 scholarship * $20,000) + (Probability of no scholarship * $0)
Expected Scholarship Money = (0.1587 * $20,000) + (0.8413 * $0)
Expected Scholarship Money = $3,174
Thus, the college would expect to disperse an average of $3,174 in merit scholarships in a given year.