Question

the heights of a group of 1000 women are normally distributed. the mean height of the group is 170 centimeters (cm) with a standard deviation of 10 cm. what is the best approximation of the number of women between 170 cm and 180 cm tall?
950
340
170
680

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.

The interval from \(\mu\) to \(\mu + \sigma\) (where \(\mu = 170\) and \(\sigma=10\), so \(170\) to \(180\)) represents half of the \(68\%\) interval (\(\mu-\sigma\) to \(\mu + \sigma\)).

Step2: Calculate the proportion and the number of women

The proportion of data from \(\mu\) to \(\mu+\sigma\) is \(\frac{68\%}{2}=34\%\).
If there are \(n = 1000\) women, then the number of women in the interval \([170,180]\) is \(n\times0.34\).

$$1000\times0.34 = 340$$

Answer:

340