Question

The average playing time of compact discs in a large collection is 32 minutes, and the standard deviation is 7 minutes.

(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations away from the mean?

1 standard deviation above the mean
1 standard deviation below the mean
2 standard deviations above the mean
2 standard deviations below the mean

(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 18 and 46 minutes? (Round the answer to the nearest whole number.)
At least %

(c) Without assuming anything about the distribution of times, what can be said about the percentage of times that are either less than 11 minutes or greater than 53 minutes? (Round the answer to the nearest whole number.)
No more than %

(d) Assuming that the distribution of times is normal, about what percentage of times are between 18 and 46 minutes? (Round the answers to two decimal places, if needed.)
%

Less than 11 min or greater than 53 min?
%

Less than 11 min?

Answer 1

The value that is 1 standard deviation above the mean is 39 minutes, and the value that is 1 standard deviation below the mean is 25 minutes. The values that are 2 standard deviations away from the mean are 46 minutes and 18 minutes. Between 18 and 46 minutes, at least 95% of the times are between 18 and 46 minutes. Assuming a normal distribution, the percentage of times that are less than 11 minutes is 0%.

Step-by-step explanation:

The value that is 1 standard deviation above the mean can be found by adding 1 standard deviation to the mean. In this case, the mean is 32 minutes and the standard deviation is 7 minutes, so 1 standard deviation above the mean is 32 + 7 = 39 minutes.

The value that is 1 standard deviation below the mean can be found by subtracting 1 standard deviation from the mean. So, 1 standard deviation below the mean is 32 - 7 = 25 minutes.

The values that are 2 standard deviations away from the mean can be found by adding or subtracting 2 standard deviations from the mean. So, 2 standard deviations above the mean is 32 + (2 * 7) = 46 minutes, and 2 standard deviations below the mean is 32 - (2 * 7) = 18 minutes.

To find the percentage of times that are between 18 and 46 minutes, we need to find the area under the normal distribution curve between these two values. The percentage can be approximated by calculating the standard normal cumulative distribution function (CDF) for each value and taking the difference. Using a standard normal table or a calculator, we find that the CDF for 18 minutes is approximately 0.025 and the CDF for 46 minutes is approximately 0.975. Subtracting these values gives us 0.975 - 0.025 = 0.95. Multiplying by 100 gives us a percentage of 95%.

Without assuming anything about the distribution of times, we can say that no more than 5% of times are either less than 11 minutes or greater than 53 minutes.

Assuming that the distribution of times is normal, we can use the standard normal CDF to find the percentage of times between 18 and 46 minutes. The CDF for 18 minutes is approximately 0.025 and the CDF for 46 minutes is approximately 0.975. Subtracting these values gives us 0.975 - 0.025 = 0.95. Multiplying by 100 gives us a percentage of 95%.

Assuming a normal distribution, the percentage of times that are less than 11 minutes can be found by calculating the CDF for 11 minutes using the standard normal distribution. Using a standard normal table or a calculator, we find that the CDF for 11 minutes is approximately 0.0. Multiplying by 100 gives us a percentage of 0%.

Answer 2

The times one and two standard deviations from the mean are 39 minutes and 25 minutes, and 46 minutes and 18 minutes respectively. Using Chebyshev's Theorem, at least 75% of times are between 18 and 46 minutes, and no more than 25% are outside of 11 and 53 minutes. Assuming normal distribution, 95.44% of times are between 18 and 46 minutes, and less than 2.28% are outside of 11 and 53 minutes.

Step-by-step explanation:

The mean of the compact discs playing time in the collection is 32 minutes, and the standard deviation is 7 minutes.

1. 1 standard deviation above the mean is mean + 1 * standard deviation = 32 + 7 = 39 minutes.
2. 1 standard deviation below the mean is mean - 1 * standard deviation = 32 - 7 = 25 minutes.
3. 2 standard deviations above the mean is mean + 2 * standard deviations = 32 + 14 = 46 minutes.
4. 2 standard deviations below the mean is mean - 2 * standard deviations = 32 - 14 = 18 minutes.

The Chebyshev's Theorem can be applied (without assuming anything about the distribution) to give us:

• (b) At least 75% of the times are between 18 and 46 minutes.
• (c) No more than 25% of the times are either less than 11 minutes or greater than 53 minutes.

Assuming a normal distribution:

• (d) About 95.44% of times are between 18 and 46 minutes (mean +/- 2 standard deviations encompass approximately 95.44% of data in a normal distribution).
• Less than 2.28% are less than 11 min or greater than 53 min (because within mean +/- 3 standard deviations, which is approximately 99.72% of data, hence outside is 100% - 99.72% = 0.28%, and since we split it to both ends, we get less than 0.28%/2 = 0.14% at each end).