Question

The test scores for a very large statistics class have a bell-shaped distribution with a mean of 70 points. If 16% of all students in the class scored above 85, what is the standard deviation of the scores? If 95% of the scores are between 60 and 80, what is the standard deviation?

Answer 1

$\mu=70 \\ P(X>85) = 0.16 \\ 1-P(X<85) = 0.16 \\ P(X<85) = 1 -0.16 = 0.84 \\ P(Z< \frac{85-70}{\sigma}) =0.84 \\ Z=0.994 \\ \frac{15}{\sigma}=0.994 \\ \sigma= 15.091$

The standard deviation of the scores is 15.091

$P(60<X<80) = 0.95 \\ P(\frac{60-70}{\sigma}<Z< \frac{80-70}{\sigma}) = 0.95 \\ P( \frac{-10}{\sigma}<Z< \frac{10}{\sigma}) = 0.95 \\ 1 -2P(Z < \frac{-10}{\sigma}) = 0.95 \\ P(Z< \frac{-10}{\sigma}) = 0.025 \\ Z= -1.96 \\ \frac{-10}{\sigma} = -1.96 \\ \sigma = 5.102$

The standard deviation is 5.102