I.use the chebyshev's theorem to find out what percentage of the values will fall between 201 and 257 for a data set with a mean of 229 and standard deviation of 14.
II. use the emperical rule to find out two values 68% of the data will fall between for a data set with a mean of 205 and standard deviation of 17. show all work
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To find the percentage of values that will fall between 201 and 257 using Chebyshev's theorem, you need to use the formula:
Percentage = 1 - (1 / k^2)
where k is a constant representing the number of standard deviations from the mean that you want to include. In this case, we want to find the percentage of values that fall between 201 and 257, which is 2 standard deviations below and above the mean of 229.
Step 1: Calculate the range:
Range = Upper Bound - Lower Bound = 257 - 201 = 56
Step 2: Calculate the number of standard deviations from the mean:
Standard Deviations = Range / Standard Deviation = 56 / 14 = 4
Step 3: Calculate the percentage using Chebyshev's theorem:
Percentage = 1 - (1 / 4^2) = 1 - (1 / 16) = 1 - 0.0625 = 0.9375 = 93.75%
Therefore, approximately 93.75% of the values will fall between 201 and 257.
Now let's use the empirical rule to find two values within which 68% of the data will fall. The empirical rule states that for a normal distribution:
- Approximately 68% of the data lies within one standard deviation of the mean.
- Approximately 95% of the data lies within two standard deviations of the mean.
- Approximately 99.7% of the data lies within three standard deviations of the mean.
Step 1: Calculate the range using one standard deviation:
Range within 1 standard deviation = Mean ± (Standard Deviation * 1) = 205 ± (17 * 1) = 205 ± 17 = 188 to 222
Step 2: Calculate the range using two standard deviations:
Range within 2 standard deviations = Mean ± (Standard Deviation * 2) = 205 ± (17 * 2) = 205 ± 34 = 171 to 239
Therefore, for a data set with a mean of 205 and a standard deviation of 17, approximately 68% of the data will fall between 188 and 222, according to the empirical rule.