Question 1199714: The scores archived by the entired population of students follow a normal distribution.
The average score for the entire population was 21.
47.72% of the students scored between 15 and 21.
Full scholarship is offered to all students with a z-score of 3.2 or more.
a. Find the standard deviation. Show your work.
b. What is the minimum score a student must achieve in order to earn the scholarship?
c. What percentage of students were offered this scholarship?
Found 2 solutions by textot, ikleyn:
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a) Find the standard deviation**
1. **Find the z-score corresponding to the 22.28th percentile:**
* Since 47.72% of students scored between 15 and 21, and the distribution is symmetric, 22.28% of students scored below 15.
* Using a standard normal distribution table or calculator, find the z-score corresponding to the 22.28th percentile.
* This z-score is approximately **-0.77**.
2. **Calculate the standard deviation:**
* Use the z-score formula:
* z = (X - μ) / σ
* where:
* z = z-score (-0.77)
* X = score (15)
* μ = mean (21)
* σ = standard deviation
* Rearrange the formula to solve for σ:
* σ = (X - μ) / z
* σ = (15 - 21) / -0.77
* σ = -6 / -0.77
* σ ≈ 7.79
**Therefore, the standard deviation of the scores is approximately 7.79.**
**b) Find the minimum score for a scholarship**
1. **Find the raw score corresponding to a z-score of 3.2:**
* Use the z-score formula:
* X = μ + z * σ
* X = 21 + 3.2 * 7.79
* X = 21 + 24.928
* X = 45.928
**Therefore, the minimum score a student must achieve to earn the scholarship is approximately 45.93.**
**c) Percentage of students offered the scholarship**
1. **Find the area to the right of z = 3.2:**
* Using a standard normal distribution table or calculator, find the area to the right of z = 3.2.
* This area represents the proportion of students with a z-score of 3.2 or more.
* This area is approximately 0.0007 or 0.07%.
**Therefore, approximately 0.07% of students were offered the scholarship.**
Answer by ikleyn(52815)  (Show Source):
You can put this solution on YOUR website! .
The scores archived by the  entire population of students follow a normal distribution.
The average score for the entire population was 21.
47.72% of the students scored between 15 and 21.
Full scholarship is offered to all students with a z-score of 3.2 or more.
a. Find the standard deviation. Show your work.
b. What is the minimum score a student must achieve in order to earn the scholarship?
c. What percentage of students were offered this scholarship?
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The solution and the answers in the post by the other tutor are incorrect in parts (a) and (b).
I came to bring correct solution to all parts (a), (b), (c).
Part (a)
- Since 47.72% of the students scored between 15 and 21,
- since 21 is the average score, and
- since a normal distribution is symmetric,
we conclude that twice more percentage of the students, 2*47.72% = 95.44% are scored between
15 and 21 + (21-15) = 21 + 6 = 27.
Now recall the "empirical rule" of the normal distribution: 95.45% of values are within 2 standard deviations.
From the data above, we conclude that 21-15 = 6 = 27 - 21 scores is two standard deviations.
Hence, one standard deviation of this normal distribution is 6/2 = 3 scores.
It is the ANSWER to question (a): the standard deviation of this normal distribution is 3 scores.
Part (a) is completed.
Part (b)
As we completed part (a), the ANSWER to question (b) is immediate: this minimum score
is 3.2 standard deviations above the average. i.e.
21 + 3.2*3 = 30.6.
It is the ANSWER to question (b): this minimum score is 30.6.
Part (c)
Question (c) asks to find the area under this normal curve on the right of the score 30.6.
The easiest way to do it is to use online free of charge calculator
https://onlinestatbook.com/2/calculators/normal_dist.html
It has very convenient and intuitively clear interface, so it, actually, does not require
any explanations. The calculator gives the value (the ANSWER)
the area = 0.0007.
It means that 0.07% of students will be offered this scholarship.
Alternatively, instead of the online calculator, you may use your regular calculator like TI-83/84
with the standard function normcdf. Then the formula to get the desired area is
z1 z2 mean, SD <<<---=== formatting pattern
area = normcdf(31.6. 9999, 21, 3).
It will give you the same result.
Solved.
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