Question

A set of data has a normal distribution with a mean of 53 and a standard deviation of 6. Find the percent of data within each interval.

1. from 47 to 65

95%, 81.5%, 68%, or 47.5%?

2. from 41 to 47?

28.5%, 54.5%, 13.5%, or 15.5%?

3. greater than 59?

16%, 13.5%, 25%, or 2.5%?

4. less than 65%?

2.5%, 16%, 84%, or 97.5%?

Answer 1

1) We need to work out two Z-scores, one for the lower number and one for the higher.

Z₁=(47-53)/6=-1; Z₂=(65-53)/6=2.

N(-1)=1-N(1)=1-0.8413=0.1587; N(2)=0.9772.

The probability is 0.9772-0.1587=0.8185 (closest answer option 81.5%).

2) Z₁=(41-53)/6=-2; Z₂=(47-53)/6=-1.

N(-2)=1-N(-2)=1-0.9772=0.0228; N(-1)=0.1587.

The probability is 0.1587-0.0228=0.1359 (closest answer option 13.5%).

3) Z=(59-53)/6=1, N(X<59)=0.8413⇒N(X>59)=1-0.8413=0.1587 (~16%).

4) <65? Z=2, N(2)=0.9772 (see Z₂ in part (1).) About 97.5%.