Consider the following.
(a) What percentage of a normal distribution is greater than the mean? (Give your answer correct to the nearest percent.)
Incorrect: Your answer is incorrect. . %
(b) What percentage is within 1 standard deviation of the mean? (Give your answer correct to the nearest percent.)
Correct: Your answer is correct. . %
(c) What percentage is greater than a value that is 1 standard deviation below the mean? (Give your answer correct to the nearest percent.)
Incorrect: Your answer is incorrect. . %
To answer these questions, we need to understand the properties of a normal distribution and how to calculate percentages based on standard deviations.
(a) What percentage of a normal distribution is greater than the mean?
In a normal distribution, the mean is the central value, and half of the distribution is on either side of the mean. Since the distribution is symmetric, exactly 50% of the values are greater than the mean, and 50% are smaller.
So, the percentage of a normal distribution that is greater than the mean is 50%.
(b) What percentage is within 1 standard deviation of the mean?
In a normal distribution, approximately 68% of the values fall within one standard deviation of the mean. To calculate this percentage, we need to find the area under the curve within one standard deviation of the mean.
To do this, we can use a standard normal distribution table or a statistical calculator to find the probability (z-score) associated with a value that is one standard deviation away from the mean.
The probability associated with a value within one standard deviation of the mean is approximately 0.6826 or 68.26%.
So, the percentage of a normal distribution within one standard deviation of the mean is approximately 68%.
(c) What percentage is greater than a value that is 1 standard deviation below the mean?
To find the percentage of a normal distribution that is greater than a value that is one standard deviation below the mean, we need to calculate the area under the curve from that value to positive infinity.
Again, we can use a standard normal distribution table or a statistical calculator to find the probability associated with a value that is one standard deviation below the mean.
The probability associated with a value that is one standard deviation below the mean is approximately 0.1587 or 15.87%.
However, since we want to find the percentage that is greater than this value, we subtract this probability from 1.
So, the percentage of a normal distribution that is greater than a value that is one standard deviation below the mean is approximately 1 - 0.1587 = 0.8413 or 84.13%.
Therefore, the correct answer to question (c) is 84%.