A probability distribution has a mean of 57 and a standard deviation of 1.4. Use Chebychev’s inequality to find the value of c that guarantees the probability is at least 96% that an outcome of the experiment lies between 57 - c and
57 - c. (Round the answer to nearest whole number.)
It would help if you proofread your questions before you posted them.
"lies between 57 - c and
57 - c" = 0
98% = mean ± Z (SD)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.48) and its Z score. Insert data into above equation and solve for Z(SD).
To use Chebyshev's inequality, we need to know the range of values within a certain number of standard deviations from the mean.
Chebyshev's inequality states that for any probability distribution, the proportion of values that lie within k standard deviations of the mean is at least 1 - 1/k^2, where k is any positive real number greater than 1.
In this case, we want to find the value of c such that the probability is at least 96% that an outcome of the experiment lies within 57 - c and 57 + c.
Let's calculate the value of c using Chebyshev's inequality.
We know that the probability is at least 96%, so the proportion of values that lie within c standard deviations of the mean is at least 1 - 1/c^2.
We can set up the following inequality:
1 - 1/c^2 ≥ 0.96
Simplifying the inequality:
1/c^2 ≤ 0.04
Taking the reciprocal of both sides:
c^2 ≥ 25
Taking the square root of both sides, and since c cannot be negative, we get:
c ≥ 5
Therefore, the value of c that guarantees the probability is at least 96% that an outcome of the experiment lies between 57 - c and 57 + c is 5.
It would help if you proofread your questions before you posted them.
"lies between 57 - c and
57 - c" = 0
98% = mean ± Z (SD)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.48) and its Z score. Insert data into above equation and solve.