Data are drawn from a symmetric and bellshaped distribution

Data are drawn from a symmetric and bell-shaped distribution with a mean of 105 and a standard deviation of 4. There are 1500 observations in the data set. a. What percentage of the observations are less than 113? (Round your answer to 1 decimal place.) Percentage of observations b. Approximately how many observations are less than 113? (Round your answer to the nearest whole number.) Number of observations

Solution

The number of observations = 1500. So we can assume that the observations are normally distributed.

Mean = 105.

Std. Dev = 4

Thus, Z-score for X = 113 will be:

Z = (113 - 105) /4

= 8 / 4

= 2

Thus, P(X < 113) = P(Z < 2) in a standard normal distribution

Looking at the value for P(Z<2) = 0.9772

Thus, 97.7% of the observations are below 113

Also,

Number of observations below 113 = (0.977 * 1500) = 1465.5 ~1466 observations

hope this helps. Ask if you have any doubts.