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.
1. Z = (score-mean)/SD
Find the Z scores and apply your theorem.
2. 99.7% fall between Z scores of ± 3.
Use the equation above.
1. Z = (s-205)/11 = 205/11 = 18.63???
This one confuses me
2. I have no idea how to do this....
Please help!!
1. Z = (183-205)/11 = -22/11 = -2
Z = (227-205)/11 = ?
Apply the theorem.
2. 3 = (score-297)/18
-3 = (score-297)/18
Solve for each score.
okay so the first would be 0.75 correct?
the second one would be
3 = -16.46
-3 = -16.46
is this correct?
Sure, let's solve these problems step by step.
I. Using Chebyshev's theorem, we can find the proportion of values that fall within a certain number of standard deviations from the mean. The theorem states that for any distribution, at least (1 - 1/k^2) of the data fall within k standard deviations from the mean, where k is any number greater than 1.
In this case, we want to find the proportion of values that fall between 183 and 227, which is a range of 44 units. To use Chebyshev's theorem, we need to find the number of standard deviations this range represents.
First, calculate the difference between the mean and the lower value of the range:
183 - 205 = -22
Next, calculate the difference between the mean and the upper value of the range:
227 - 205 = 22
Since the standard deviation is given as 11, we can determine the number of standard deviations by dividing the differences by the standard deviation:
-22 / 11 = -2
22 / 11 = 2
Now, we have the values of -2 and 2, which represent the number of standard deviations from the mean.
Using Chebyshev's theorem, at least (1 - 1/k^2) of the data fall within k standard deviations from the mean. So, for k = 2, at least (1 - 1/2^2) = (1 - 1/4) = 3/4 = 0.75, or 75% of the data will fall within the range of 183 and 227.
Therefore, 75% of the values will fall between 183 and 227 for this data set.
II. The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to data distributions that are bell-shaped and follow a normal distribution. It states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
In this case, we want to find the range that includes 99.7% of the data.
Given a mean of 297 and a standard deviation of 18, we can use the Empirical Rule as follows:
First, calculate one standard deviation:
1 * 18 = 18
Next, calculate two standard deviations:
2 * 18 = 36
Finally, calculate three standard deviations:
3 * 18 = 54
So, 99.7% of the data will fall between the range of (297 - 54) to (297 + 54), which is (243, 351).
Therefore, 99.7% of the data will fall between the values of 243 and 351 for this data set.