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 95 95 34 34 47

Answer 1

To determine how many data points Mona should expect to find between 1000 and 1800 in a normally distributed dataset with parameters \(N(1400, 200)\), we need to first calculate the z-scores for the values 1000 and 1800.

The formula for the z-score is:

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

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

First, let's calculate the z-scores.

For \(X = 1000\):

\[
z_{1000} = \frac{(1000 - 1400)}{200} = \frac{-400}{200} = -2
\]

For \(X = 1800\):

\[
z_{1800} = \frac{(1800 - 1400)}{200} = \frac{400}{200} = 2
\]

Next, we will look up these z-scores in the standard normal distribution table, which gives us the probability of a value lying below a certain z-score.

- The cumulative probability for \(z = -2\) is approximately 0.0228.
- The cumulative probability for \(z = 2\) is approximately 0.9772.

Now, to find the probability of a data point falling between 1000 and 1800, we subtract the cumulative probability for \(z = -2\) from that of \(z = 2\):

\[
P(1000 < X < 1800) = P(Z < 2) - P(Z < -2) \approx 0.9772 - 0.0228 = 0.9544
\]

Now, we can use this probability to estimate the number of data points in the sample of 50.

\[
\text{Expected number of data points} = 0.9544 \times 50 \approx 47.72
\]

Since we are looking for a whole number, we round this to about 48.

Thus, the closest option to 48 in the options provided is **47**.

So the answer is **47**.