Question
Mona is given the 1400 for the Mean and 200 for the Standard Deviation 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
47
47
17
17
34
34
95
Responses
47
47
17
17
34
34
95
Answers
GPT-4o mini
To find out how many data points Mona should expect to find between 1000 and 1800, we can use the properties of the normal distribution.
1. We know that the mean (μ) is 1400 and the standard deviation (σ) is 200.
2. We need to calculate the z-scores for 1000 and 1800.
For 1000:
\[
z = \frac{(X - \mu)}{\sigma} = \frac{(1000 - 1400)}{200} = \frac{-400}{200} = -2
\]
For 1800:
\[
z = \frac{(X - \mu)}{\sigma} = \frac{(1800 - 1400)}{200} = \frac{400}{200} = 2
\]
3. Now we can look up the z-scores in the standard normal distribution table or use the empirical rule:
- A z-score of -2 corresponds to the percentile of about 2.5% (the area to the left of the z-score).
- A z-score of 2 corresponds to the percentile of about 97.5% (again, the area to the left of the z-score).
4. The area between z = -2 and z = 2 represents the percentage of data points between 1000 and 1800:
\[
97.5\% - 2.5\% = 95\%
\]
5. Since there are 50 data points in total, we calculate:
\[
0.95 \times 50 = 47.5
\]
Since we are dealing with data points, we can round this to approximately 47.
Thus, the expected number of data points between 1000 and 1800 is approximately **47**.
1. We know that the mean (μ) is 1400 and the standard deviation (σ) is 200.
2. We need to calculate the z-scores for 1000 and 1800.
For 1000:
\[
z = \frac{(X - \mu)}{\sigma} = \frac{(1000 - 1400)}{200} = \frac{-400}{200} = -2
\]
For 1800:
\[
z = \frac{(X - \mu)}{\sigma} = \frac{(1800 - 1400)}{200} = \frac{400}{200} = 2
\]
3. Now we can look up the z-scores in the standard normal distribution table or use the empirical rule:
- A z-score of -2 corresponds to the percentile of about 2.5% (the area to the left of the z-score).
- A z-score of 2 corresponds to the percentile of about 97.5% (again, the area to the left of the z-score).
4. The area between z = -2 and z = 2 represents the percentage of data points between 1000 and 1800:
\[
97.5\% - 2.5\% = 95\%
\]
5. Since there are 50 data points in total, we calculate:
\[
0.95 \times 50 = 47.5
\]
Since we are dealing with data points, we can round this to approximately 47.
Thus, the expected number of data points between 1000 and 1800 is approximately **47**.