A large population has a mean of 200 and a standard deviation of 40. Which one of the following best describes the percentage of measurements that fall in an interval that is within two standard deviations of the mean?
What are your alternatives?
Approximately 95% for any normal distribution.
To find the percentage of measurements that fall within two standard deviations of the mean, we can refer to the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule:
- Approximately 68% of measurements fall within one standard deviation of the mean.
- Approximately 95% of measurements fall within two standard deviations of the mean.
- Approximately 99.7% of measurements fall within three standard deviations of the mean.
Since we are interested in the percentage within two standard deviations of the mean, the best description would be "approximately 95%".
To find the percentage of measurements that fall within two standard deviations of the mean, we can use the empirical rule, also known as the 68-95-99.7 rule. According to this rule, for a normal distribution:
- Approximately 68% of the measurements fall within one standard deviation of the mean.
- Approximately 95% of the measurements fall within two standard deviations of the mean.
- Approximately 99.7% of the measurements fall within three standard deviations of the mean.
In this case, the mean is 200 and the standard deviation is 40. To find the interval within two standard deviations of the mean, we calculate the lower and upper limits as follows:
Lower limit = mean - (2 * standard deviation) = 200 - (2 * 40) = 120
Upper limit = mean + (2 * standard deviation) = 200 + (2 * 40) = 280
So, the measurements falling within this interval (120 to 280) are considered to be within two standard deviations of the mean. Therefore, based on the empirical rule, approximately 95% of the measurements will fall within this interval.