I. Use Chebyshev’s theorem to find what percent of the values will fall between 156 and 276 for a data set with a mean of 216 and standard deviation of 12.
II. Use the Empirical Rule to find what two values 68% of the data will fall between for a data set with a mean of 219 and standard deviation of 20.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
I. To use Chebyshev's theorem to find the percent of values that will fall between 156 and 276, we start by calculating the number of standard deviations each boundary is away from the mean.
The first step is to calculate the difference between each boundary and the mean:
Upper boundary: 276 - 216 = 60
Lower boundary: 156 - 216 = -60
Next, we divide these differences by the standard deviation:
Upper boundary: 60 / 12 = 5
Lower boundary: -60 / 12 = -5
Thus, we see that the upper boundary is 5 standard deviations away from the mean, and the lower boundary is -5 standard deviations away from the mean.
According to Chebyshev's theorem, for any data set, at least (1 - 1/k^2) of the values will fall within k standard deviations of the mean, where k is any positive constant greater than 1.
In this case, we need to find the percent of values that fall between -5 and 5 standard deviations.
To calculate this, we use the formula: (1 - 1/k^2) * 100
For k = 5, we have:
(1 - 1/5^2) * 100 = (1 - 1/25) * 100 = (24/25) * 100 = 96
Therefore, at least 96% of the values will fall between 156 and 276 for this data set.
II. To use the Empirical Rule to find the two values between which 68% of the data will fall, we need to consider the z-scores.
The Empirical Rule states that for a data set with a bell-shaped distribution, approximately 68% of the data will fall within one standard deviation of the mean, approximately 95% of the data will fall within two standard deviations of the mean, and approximately 99.7% will fall within three standard deviations of the mean.
In this case, we have a mean of 219 and a standard deviation of 20.
To find the range of two values between which 68% of the data will fall, we start by calculating one standard deviation.
One standard deviation: 20 * 1 = 20
Next, we calculate the two values by adding and subtracting one standard deviation from the mean:
Upper value: 219 + 20 = 239
Lower value: 219 - 20 = 199
Therefore, 68% of the data will fall between 199 and 239 for this data set according to the Empirical Rule.