The mean playing time for a large collection of music CDs is 37 minutes, and the standard deviation is
7 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 23 and 51 minutes?
Approximately what percentage of times are less than 16 min or greater than 58 min?
Approximately what percentage of times are less than 16 min?
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(a)
1 standard deviation above the mean: mean + standard deviation = 37 + 7 = 44 min
1 standard deviation below the mean: mean - standard deviation = 37 - 7 = 30 min
2 standard deviations above the mean: mean + 2 * standard deviation = 37 + 2 * 7 = 51 min
2 standard deviations below the mean: mean - 2 * standard deviation = 37 - 2 * 7 = 23 min
1 standard deviation above the mean: 44 min
1 standard deviation below the mean: 30 min
2 standard deviations above the mean: 51 min
2 standard deviations below the mean: 23 min
(b)
To find the percentage of times between 23 and 51 minutes, we need to find the area under the curve between these two points. Given that the distribution is approximately symmetric and mound-shaped, we can estimate this percentage using the empirical rule. According to the empirical rule, approximately 68% of the data falls within 1 standard deviation of the mean, and approximately 95% falls within 2 standard deviations of the mean. Therefore, we can estimate that approximately 68% - 95% = 27% of the data falls between 1 and 2 standard deviations away from the mean.
To find the percentage of times less than 16 min or greater than 58 min, we can again use the empirical rule. Approximately 95% falls within 2 standard deviations of the mean, so approximately 5% falls outside of this range. Therefore, we can estimate that approximately 5% of the data is less than 16 min or greater than 58 min.
To find the percentage of times less than 16 min, we can estimate that approximately 2.5% (half of 5%) of the data falls below 2 standard deviations below the mean.
Approximately what percentage of times are between 23 and 51 minutes?
Approximately 27% of times are between 23 and 51 minutes.
Approximately what percentage of times are less than 16 min or greater than 58 min?
Approximately 5% of times are less than 16 min or greater than 58 min.
Approximately what percentage of times are less than 16 min?
Approximately 2.5% of times are less than 16 min.
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