Question

Mona 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)
Responses

17
17

34
34

95
95

47

Answer 1

To find out how many data points Mona should expect to find between 1000 and 1800 in a normal distribution \( N(1400, 200) \), we first need to standardize the values using the z-score formula:

\[
z = \frac{(X - \mu)}{\sigma}
\]

Where:
- \( X \) is the value we are interested in,
- \( \mu \) is the mean (1400),
- \( \sigma \) is the standard deviation (200).

1. For \( X = 1000 \):
\[
z = \frac{(1000 - 1400)}{200} = \frac{-400}{200} = -2
\]

2. For \( X = 1800 \):
\[
z = \frac{(1800 - 1400)}{200} = \frac{400}{200} = 2
\]

Next, we look up the z-scores in the standard normal distribution table (or use a calculator):

- The area to the left of \( z = -2 \) is approximately 0.0228 (or 2.28%).
- The area to the left of \( z = 2 \) is approximately 0.9772 (or 97.72%).

To find the area (or the proportion of data points) between z = -2 and z = 2:
\[
P(-2 < Z < 2) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
\]

Now, since Mona has 50 data points, we can calculate the expected number of data points between 1000 and 1800:
\[
0.9544 \times 50 \approx 47.22
\]

Rounding to the nearest whole number, we find that Mona should expect about **47 data points** between 1000 and 1800.

So, the correct response is **47**.