Question
A set of data has a normal distribution with a mean of 46 and a standard deviation of 9. Find the percent of data within the following interval.
from 37 to 55
from 37 to 55
Answers
GPT 3.5
To find the percent of data within the interval from 37 to 55 in a normal distribution with a mean of 46 and a standard deviation of 9, we first need to find the z-scores corresponding to the two values.
For 37:
z = (37 - 46) / 9 = -1
For 55:
z = (55 - 46) / 9 = 1
Next, we find the area under the normal curve between these two z-scores. We can do this by finding the area to the left of z = 1 (which is 0.8413) and subtracting the area to the left of z = -1 (which is 0.1587).
Therefore, the area between z = -1 and z = 1 is 0.8413 - 0.1587 = 0.6826.
To find the percent of data within the interval from 37 to 55, we multiply this by 100:
0.6826 * 100 = 68.26%
So, approximately 68.26% of the data falls within the interval from 37 to 55 in a normal distribution with a mean of 46 and a standard deviation of 9.
For 37:
z = (37 - 46) / 9 = -1
For 55:
z = (55 - 46) / 9 = 1
Next, we find the area under the normal curve between these two z-scores. We can do this by finding the area to the left of z = 1 (which is 0.8413) and subtracting the area to the left of z = -1 (which is 0.1587).
Therefore, the area between z = -1 and z = 1 is 0.8413 - 0.1587 = 0.6826.
To find the percent of data within the interval from 37 to 55, we multiply this by 100:
0.6826 * 100 = 68.26%
So, approximately 68.26% of the data falls within the interval from 37 to 55 in a normal distribution with a mean of 46 and a standard deviation of 9.