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.