Question

1. 2000 freshmen at state university took a biology test. the scores were distributed normally with a mean of 70 and a standard deviation of 5. 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 65 and 75?
b) what percentage of scores are between scores 60 and 70?
c) what percentage of scores are between scores 60 and 85?
d) what percentage of scores is less than a score of 55?
e) what percentage of scores is greater than a score of 80?
f) approximately how many biology students scored between 60 and 70?
g) approximately how many biology students scored between 55 and 60?

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\mu\pm\sigma\)) of the mean.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\mu\pm2\sigma\)) of the mean.
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\mu\pm3\sigma\)) of the mean.

Given \(\mu = 70\) and \(\sigma=5\)

Step2: Solve part a)

For scores between \(65\) and \(75\):
\(65 = 70 - 5=\mu-\sigma\) and \(75=70 + 5=\mu+\sigma\)
By the empirical rule, the percentage of scores between \(\mu-\sigma\) and \(\mu+\sigma\) is \(68\%\)

Step3: Solve part b)

For scores between \(60\) and \(70\):
\(60 = 70- 2\times5=\mu - 2\sigma\)
The percentage of scores between \(\mu - 2\sigma\) and \(\mu\) is \(\frac{95\%}{2}=47.5\%\)

Step4: Solve part c)

For scores between \(60\) and \(85\):
\(60=\mu - 2\sigma\) and \(85=70+3\times5=\mu + 3\sigma\)
The percentage of scores between \(\mu - 2\sigma\) and \(\mu+3\sigma\) is \(47.5\%+34\%+13.5\%+2.35\% = 97.35\%\)

Step5: Solve part d)

For scores less than \(55\):
\(55=\mu-3\sigma\)
The percentage of scores less than \(\mu - 3\sigma\) is \(0.15\%\)

Step6: Solve part e)

For scores greater than \(80\):
\(80=70 + 2\times5=\mu+2\sigma\)
The percentage of scores greater than \(\mu + 2\sigma\) is \(\frac{100 - 95}{2}\%=2.5\%\)

Step7: Solve part f)

For scores between \(60\) and \(70\), percentage is \(47.5\%\)
Number of students \(=0.475\times2000 = 950\)

Step8: Solve part g)

For scores between \(55\) and \(60\), percentage is \(2.35\%\)
Number of students \(=0.0235\times2000=47\)

Answer:

a) \(68\%\)
b) \(47.5\%\)
c) \(97.35\%\)
d) \(0.15\%\)
e) \(2.5\%\)
f) \(950\)
g) \(47\)