Suppose that the blood pressure of the human inhabitants of a certain Pacific island is distributed with mean, \mu = 108 mmHg and standard deviation , \sigma = 6 mmHg. According to Chebyshev's Theorem, at least what percentage of the islanders have blood pressure in the range from 93.6 mmHg to 122.4 mmHg ?

I have the answer, but I don't understand how to get it...please help!

82.64%

Don't know about theorem, but here is method.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to your Z scores. Multiply by 100.

To use Chebyshev's Theorem to calculate the percentage of islanders with blood pressure in the range from 93.6 mmHg to 122.4 mmHg, follow these steps:

Step 1: Determine the range between the lower and upper bounds of the blood pressure range: 122.4 mmHg - 93.6 mmHg = 28.8 mmHg.

Step 2: Find the number of standard deviations the range is from the mean by dividing the range by the standard deviation: 28.8 mmHg / 6 mmHg = 4.8.

Step 3: Use Chebyshev's Theorem, which states that for any number k greater than 1, the proportion of any data set lying within k standard deviations of the mean is at least (1 - 1/k^2).

In this case, k = 4.8. Plug in this value into the formula to get (1 - 1/4.8^2).

Step 4: Calculate the minimum percentage using the formula: (1 - 1/4.8^2) x 100%.

This will give you the minimum percentage of islanders with blood pressure in the range from 93.6 mmHg to 122.4 mmHg.

To use Chebyshev's Theorem to find the percentage of islanders with blood pressure in the given range, we need to calculate the probability within a certain number of standard deviations from the mean. Chebyshev's Theorem states that for any random variable, at least (1 - 1/k^2) of the data falls within k standard deviations from the mean, where k is any positive number greater than 1.

In this case, we want to find the percentage of islanders with blood pressure in the range from 93.6 mmHg to 122.4 mmHg.

First, we need to calculate the number of standard deviations away from the mean that each end point of the range is:

Z1 = (93.6 - 108) / 6 = -2.4
Z2 = (122.4 - 108) / 6 = 2.4

Now, we can find the probability within 2.4 standard deviations from the mean using Chebyshev's Theorem:

P(|Z| < 2.4) = 1 - (1 / 2.4^2) = 1 - (1 / 5.76) = 1 - 0.1736 = 0.8264

So, at least 82.64% of the islanders have blood pressure in the range from 93.6 mmHg to 122.4 mmHg according to Chebyshev's Theorem.

It's important to note that Chebyshev's Theorem provides a lower bound on the probability and gives a wide range of values. It does not account for the distribution shape or any specific characteristics of the data.