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.
Obviously, this is not a repost as I only posted this question once. So if someone else posted the same question, I have NO control over what they do.
I. To find the percentage of values that fall between 156 and 276 using Chebyshev's theorem, we need to calculate the range within a certain number of standard deviations from the mean.
1. First, we calculate the range:
Range = 276 - 156 = 120
2. Next, we determine the number of standard deviations from the mean that covers this range:
Number of standard deviations = Range / Standard deviation
= 120 / 12
= 10
3. According to Chebyshev's theorem, at least (1 - 1/k^2) of the values will fall within k standard deviations from the mean, where k is the number of standard deviations.
In this case, k = 10, so the proportion of values falling within 10 standard deviations from the mean is at least (1 - 1/10^2) = 0.99.
4. To convert this proportion to percentages, we multiply by 100:
Percentage of values falling within 10 standard deviations = 0.99 * 100 = 99%.
Therefore, at least 99% of the values will fall between 156 and 276 for the given data set.
II. The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normally distributed data set:
- Approximately 68% of the values will fall within one standard deviation of the mean.
- Approximately 95% of the values will fall within two standard deviations of the mean.
- Approximately 99.7% of the values will fall within three standard deviations of the mean.
In this case, we want to find the two values between which 68% of the data will fall, given a mean of 219 and standard deviation of 20.
1. First, we calculate one standard deviation:
One standard deviation = Standard deviation = 20
2. Next, we calculate the range within one standard deviation from the mean:
Lower value = Mean - One standard deviation = 219 - 20 = 199
Upper value = Mean + One standard deviation = 219 + 20 = 239
So, 68% of the data will fall between 199 and 239.