Question

500 freshmen at schaumburg high school took an algebra test. the scores were distributed normally with a mean of 75 and a standard deviation of 7. label the mean and three standard deviations from the mean.
answer the following questions based on the data:
a) what percentage of scores are between scores 61 and 82?
b) what percentage of scores are between scores 75 and 82?
c) what percentage of scores are between scores 61 and 89?
d) what percentage of scores is less than a score of 61?
e) what percentage of scores is greater than a score of 96?
f) approximately how many algebra students scored between 61 and 89?
g) approximately how many algebra students scored between 68 and 82?
h) approximately how many algebra students scored between 61 and 75?
i) approximately how many algebra students scored between 89 and 96?
j) approximately how many algebra students scored higher than 89?

Explanation:

a)

Step1: Identify the percentages for the intervals

The score \(61\) is \(\mu - 2\sigma\) (\(75-2\times7 = 61\)) and \(82\) is \(\mu+\sigma\) (\(75 + 7=82\)).
The percentage between \(\mu - 2\sigma\) and \(\mu-\sigma\) is \(13.5\%\), between \(\mu-\sigma\) and \(\mu\) is \(34\%\), and between \(\mu\) and \(\mu+\sigma\) is \(34\%\).

Step2: Sum the percentages

\(13.5\%+34\% + 34\%=81.5\%\)

b)

Step1: Identify the interval

The interval is between \(\mu\) (\(75\)) and \(\mu+\sigma\) (\(82\)).

Step2: Recall the percentage

The percentage between \(\mu\) and \(\mu+\sigma\) in a normal distribution (using the empirical rule) is \(34\%\)

c)

Step1: Identify the intervals

\(61=\mu - 2\sigma\), \(89=\mu + 2\sigma\)

Step2: Sum the percentages

The percentage between \(\mu - 2\sigma\) and \(\mu-\sigma\) is \(13.5\%\), between \(\mu-\sigma\) and \(\mu\) is \(34\%\), between \(\mu\) and \(\mu+\sigma\) is \(34\%\), and between \(\mu+\sigma\) and \(\mu + 2\sigma\) is \(13.5\%\)
\(13.5\%+34\%+34\%+13.5\% = 95\%\)

d)

Step1: Identify the position

\(61=\mu - 2\sigma\)

Step2: Recall the percentage

The percentage less than \(\mu - 2\sigma\) is \(2.35\%\)

e)

Step1: Identify the position

\(96=\mu+3\sigma\)

Step2: Recall the percentage

The percentage greater than \(\mu + 3\sigma\) is \(0.15\%\)

f)

Step1: Use the percentage from part c

From part c, the percentage between \(61\) and \(89\) is \(95\%\)

Step2: Calculate the number of students

\(n = 0.95\times500=475\)

g)

Step1: Identify the intervals

\(68=\mu-\sigma\), \(82=\mu+\sigma\)

Step2: Sum the percentages

The percentage between \(\mu-\sigma\) and \(\mu\) is \(34\%\), and between \(\mu\) and \(\mu+\sigma\) is \(34\%\)
\(n=(0.34 + 0.34)\times500=0.68\times500 = 340\)

h)

Step1: Identify the intervals

\(61=\mu - 2\sigma\), \(75=\mu\)

Step2: Sum the percentages

The percentage between \(\mu - 2\sigma\) and \(\mu-\sigma\) is \(13.5\%\), and between \(\mu-\sigma\) and \(\mu\) is \(34\%\)
\(n=(0.135+0.34)\times500=0.475\times500 = 237.5\approx238\)

i)

Step1: Identify the intervals

\(89=\mu + 2\sigma\), \(96=\mu+3\sigma\)

Step2: Calculate the percentage

The percentage between \(\mu + 2\sigma\) and \(\mu+3\sigma\) is \(2.35\%\)
\(n=0.0235\times500 = 11.75\approx12\)

j)

Step1: Identify the position

Scores higher than \(89=\mu + 2\sigma\)

Step2: Sum the percentages

The percentage greater than \(\mu + 2\sigma\) is \(2.35\%+0.15\%=2.5\%\)
\(n=0.025\times500=12.5\approx13\)

Answer:

a) \(81.5\%\)
b) \(34\%\)
c) \(95\%\)
d) \(2.35\%\)
e) \(0.15\%\)
f) \(475\)
g) \(340\)
h) \(238\)
i) \(12\)
j) \(13\)