Question

mara is given the model ( n(1400,200) ) for a set of 50 data and wants to make some predictions. about how many data points should she expect to find between 1000 and 1800? (1 point)
34
47
11
45

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution \(N(\mu,\sigma)\), about \(68\%\) of the data lies within \(\mu\pm\sigma\), about \(95\%\) lies within \(\mu\pm2\sigma\), and about \(99.7\%\) lies within \(\mu\pm3\sigma\). Here, \(\mu = 1400\) and \(\sigma=200\). Calculate the number of standard - deviations for the bounds:
For \(x_1 = 1000\), \(z_1=\frac{1000 - 1400}{200}=\frac{- 400}{200}=-2\)
For \(x_2 = 1800\), \(z_2=\frac{1800 - 1400}{200}=\frac{400}{200}=2\)

Step2: Find the proportion of data between \(z=-2\) and \(z = 2\)

According to the empirical rule, the proportion of data within \(z=-2\) and \(z = 2\) is \(p = 0.95\)

Step3: Calculate the number of data points

We have a total of \(n = 50\) data points. The number of data points \(N\) between \(1000\) and \(1800\) is \(N=n\times p\). Substitute \(n = 50\) and \(p=0.95\) into the formula: \(N=50\times0.95 = 47.5\approx47\)

Answer:

\(47\)