The mean playing time for a large collection of music CDs is 32 minutes, and the standard deviation is 2 minutes.
(a) What value (in minutes) is 1 standard deviation above the mean? One standard deviation below the mean? What values are 2 standard deviations away from the mean?
1 standard deviation above the mean ____ min
1 standard deviation below the mean ____ min
2 standard deviations above the mean _____ min
2 standard deviations below the mean _____ min
(b) Assume that the distribution of times is mound shaped and approximately symmetric.
Approximately what percentage of times are between 28 and 36 minutes?
Approximately what percentage of times are less than 26 min or greater than 38 min?
Approximately what percentage of times are less than 26 min?
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(a)
1 standard deviation above the mean = mean + 1 standard deviation = 32 + 2 = 34 minutes
1 standard deviation below the mean = mean - 1 standard deviation = 32 - 2 = 30 minutes
2 standard deviations above the mean = mean + 2 standard deviations = 32 + (2 * 2) = 36 minutes
2 standard deviations below the mean = mean - 2 standard deviations = 32 - (2 * 2) = 28 minutes
1 standard deviation above the mean: 34 min
1 standard deviation below the mean: 30 min
2 standard deviations above the mean: 36 min
2 standard deviations below the mean: 28 min
(b)
To find the percentage of times between 28 and 36 minutes, we need to calculate the area under the curve between these two values. Since the distribution is mound-shaped and approximately symmetric, we can use the empirical rule.
The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
So, the percentage of times between 28 and 36 minutes is approximately 68%.
To find the percentage of times less than 26 minutes or greater than 38 minutes, we need to calculate the area under the curve to the left of 26 minutes and the area under the curve to the right of 38 minutes. Again, using the empirical rule:
Approximately 95% of the times fall within two standard deviations of the mean. This means that approximately 2.5% of the times are more than two standard deviations (greater than 4 minutes) below the mean and 2.5% of the times are more than two standard deviations (greater than 4 minutes) above the mean.
Therefore, the percentage of times less than 26 minutes or greater than 38 minutes is approximately 2.5% + 2.5% = 5%.
To find the percentage of times less than 26 minutes, we need to calculate the area under the curve to the left of 26 minutes. According to the empirical rule, approximately 2.5% of the times fall more than two standard deviations (4 minutes) below the mean. Therefore, the percentage of times less than 26 minutes is approximately 2.5%.
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