I. Use Chebyshev’s theorem to find what percent of the values will fall between 183 and 227 for a data set with a mean of 205 and standard deviation of 11.
II. Use the Empirical Rule to find what two values 99.7% of the data will fall between for a data set with a mean of 297 and standard deviation of 18.
I. To find the percent of values that will fall between 183 and 227 using Chebyshev's theorem, we first need to calculate the range or interval around the mean that we're interested in.
Chebyshev's theorem states that for any data set, regardless of its shape, at least (1 - 1/k^2) * 100% of the values will fall within k standard deviations of the mean. Here, k is any positive number greater than 1.
In our case, we want to find the percentage of values that will fall between 183 and 227, which is a range of 44 units. To calculate the number of standard deviations that correspond to this range, we divide the range by the standard deviation:
Number of standard deviations = Range / Standard deviation = 44 / 11 = 4.
According to Chebyshev's theorem, at least (1 - 1/4^2) * 100% = 75% of the values will fall within 4 standard deviations of the mean. This means that at least 75% of the values will fall between 183 and 227.
II. The Empirical Rule, also known as the 68-95-99.7 Rule, can be used to determine the range of values that a certain percentage of data will fall between. The rule states that, for a data set that follows a normal distribution:
- Approximately 68% of the values will fall within 1 standard deviation of the mean.
- Approximately 95% of the values will fall within 2 standard deviations of the mean.
- Approximately 99.7% of the values will fall within 3 standard deviations of the mean.
In this case, we want to find the two values that 99.7% of the data will fall between. This means we need to find the range that is three standard deviations away from the mean.
To calculate the range, we multiply the standard deviation by 3: Range = Standard deviation * 3 = 18 * 3 = 54.
Therefore, 99.7% of the data will fall between the mean minus 54 (297 - 54 = 243) and the mean plus 54 (297 + 54 = 351).
In summary:
- Using Chebyshev's theorem, at least 75% of the values will fall between 183 and 227.
- Using the Empirical Rule, 99.7% of the data will fall between 243 and 351.