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 find the answers to these questions, we need to use the properties of the normal distribution.
(a) What percentage of a normal distribution is greater than the mean?
In a standard normal distribution, i.e., a normal distribution with a mean of 0 and a standard deviation of 1, the area to the right of the mean is 50%. This means that the percentage of a normal distribution greater than the mean is approximately 50%.
To find the answer for a normal distribution with a specific mean and standard deviation, we can use the Z-score formula and the Z-table. The Z-score formula is Z = (x - μ) / σ, where Z is the standard score (Z-score), x is the value in question, μ is the mean, and σ is the standard deviation.
To find the Z-score at the mean (Z = 0), we can use the formula Z = (mean - mean) / standard deviation, which simplifies to Z = 0.
Therefore, the percentage of a normal distribution greater than the mean is approximately 50%.
(b) What percentage is within 1 standard deviation of the mean?
In a standard normal distribution, approximately 68% of the data falls within 1 standard deviation of the mean. This means that if we have a normal distribution with a specific mean and standard deviation, approximately 68% of the data will fall within mean ± 1 standard deviation.
(c) What percentage is greater than a value that is 1 standard deviation below the mean?
To find the answer to this question, we can use the Z-score formula mentioned earlier. We need to calculate the Z-score for a value that is 1 standard deviation below the mean, which means Z = (mean - (mean - standard deviation)) / standard deviation.
Simplifying this equation, we get Z = 1.
To find the percentage of a normal distribution greater than this Z-score, we can use a standard normal distribution table or calculator. The value of Z = 1 corresponds to approximately 84.13% in the table.
Therefore, the percentage of a normal distribution greater than a value that is 1 standard deviation below the mean is approximately 84%.