Question

Data with 250 observations are drawn from a bell-shaped distribution with a mean of 50 and a standard deviation of 12. Approximately how many observations are more than 74?

Answer 1

Answer:

Approximately 6 observations are more than 74

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 50, \sigma = 12$

Approximately how many observations are more than 74?

First step is finding the percentage of observations which are higher than 74, which is 1 subtracted by the pvalue of Z when X = 74. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{74 - 50}{12}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

1 - 0.9772 = 0.0228

Out of 250

0.0228*250 = 5.7

Approximately 6 observations are more than 74