Question
A set of data has a normal distribution with a mean of 51 and a standard deviation of 8. Find the percent of data within the following interval.
from 43 to 59
from 43 to 59
Answers
GPT 3.5
To find the percent of data within the interval from 43 to 59, we need to first calculate the z-scores for each of the values.
For x = 43:
z = (43 - 51) / 8 = -1
For x = 59:
z = (59 - 51) / 8 = 1
Next, we look up the corresponding z-scores in a standard normal distribution table:
- The z-score of -1 corresponds to an area of 0.1587 (or 15.87%)
- The z-score of 1 corresponds to an area of 0.8413 (or 84.13%)
To find the percent of data within the interval from 43 to 59, we subtract the area to the left of 43 from the area to the left of 59:
P(43 < x < 59) = P(x < 59) - P(x < 43)
= 0.8413 - 0.1587
= 0.6826
So, approximately 68.26% of the data falls within the interval from 43 to 59.
For x = 43:
z = (43 - 51) / 8 = -1
For x = 59:
z = (59 - 51) / 8 = 1
Next, we look up the corresponding z-scores in a standard normal distribution table:
- The z-score of -1 corresponds to an area of 0.1587 (or 15.87%)
- The z-score of 1 corresponds to an area of 0.8413 (or 84.13%)
To find the percent of data within the interval from 43 to 59, we subtract the area to the left of 43 from the area to the left of 59:
P(43 < x < 59) = P(x < 59) - P(x < 43)
= 0.8413 - 0.1587
= 0.6826
So, approximately 68.26% of the data falls within the interval from 43 to 59.