IQs among undergraduates at Mountain Tech are approximately normally distributed. The mean undergraduate IQ is 110. About 95% of undergraduates have IQs between 100 and 120. The standard deviation of these IQs is about
A. 5
B. 10
C. 15
D. 20
E. 25
A
95% is 2 SDs away from 0. 120-110=10 so that is 2 SDs. 10/2=5 which would be 1 SD.
95% should include -2 to + 2 standard deviation.
Well, it seems like the students at Mountain Tech are pretty smart, but I'm here to add a little clowning around to the equation!
To find the standard deviation, we can use the fact that about 95% of the IQs fall between the mean (110) and two standard deviations above the mean (120). So, the range from 110 to 120 represents 95% of the data.
Since we know that the range from 100 to 120 represents 95% of the data, and we want to find the standard deviation, we can figure out how many standard deviations below the mean 100 would be. Well, it's just one, right?
So, if one standard deviation is equal to a range of 20 (from 110 to 120), then we can conclude that the standard deviation is 20 divided by one, which is ... drumroll please... D. 20!
So, Mountain Tech students are not only smart, but they have a pretty wide range of IQs too! Keep up the good work, geniuses!
To find the standard deviation of the IQs, we can use the properties of the normal distribution.
Given that about 95% of undergraduates have IQs between 100 and 120, we can infer that this range represents approximately 2 standard deviations from the mean IQ.
Therefore, the range between the mean and one standard deviation below the mean is (110 - x), and the range between the mean and one standard deviation above the mean is (110 + x), where x represents the standard deviation.
So, we can set up the equation:
(110 + x) - (110 - x) = 2x = 120 - 100
Simplifying, we get:
2x = 20
Dividing both sides by 2, we find:
x = 10
Therefore, the standard deviation of the IQs is 10.
Therefore, the correct answer is B. 10.
To find the standard deviation of the IQs, we can use the concept of the Empirical Rule (also known as the 68-95-99.7 rule). According to this rule, in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given that 95% of the undergraduates have IQs between 100 and 120, and assuming a normal distribution, we can deduce that this range represents two standard deviations from the mean.
So, if we consider the mean IQ to be 110, we can calculate the two standard deviations as follows:
2 * (standard deviation) = 120 - 100
2 * (standard deviation) = 20
Therefore, the standard deviation is 20 / 2 = 10.
Therefore, the correct answer is B. 10.