Question

A data set with a mean of 34 and a standard deviation of 2.5 is normally distributed

According to the Empirical Rule, what percent of the data is in each of the following ranges? Round to the nearest tenth of a percent if necessary.
Between
34 and 39
Less than
31.5
Between
29 and 36.5
Percentage
%
%

Answer 1

a)
`z= (34-34)/(2.5)= 0`

`z= (39-34)/(2.5)= 2`

And we want the probability from 0 to two deviations above the mean and we got 95/2 = 47.5 %

b)
`P(X<31.5)`

`z= (31.5-34)/(2.5)= -1`

So one deviation below the mean we have: (100-68)/2 = 16%

c)
`z= (29-34)/(2.5)= -2`

`z= (36.5-34)/(2.5)= 1`

For this case below 2 deviation from the mean we have 2.5% and above 1 deviation from the mean we got 16% and then the percentage between -2 and 1 deviation above the mean we got: (100-16-2.5)% = 81.5%

Explanation:

For this case we have a random variable with the following parameters:

`X \sim N(\mu = 34, \sigma=2.5)`

From the empirical rule we know that within one deviation from the mean we have 68% of the values, within two deviations we have 95% and within 3 deviations we have 99.7% of the data.

We want to find the following probability:

`P(34 < X<39)`

We can find the number of deviation from the mean with the z score formula:

`z= (X -\mu)/(\sigma)`

And replacing we got

`z= (34-34)/(2.5)= 0`

`z= (39-34)/(2.5)= 2`

And we want the probability from 0 to two deviations above the mean and we got 95/2 = 47.5 %

For the second case:

`P(X<31.5)`

`z= (31.5-34)/(2.5)= -1`

So one deviation below the mean we have: (100-68)/2 = 16%

For the third case:

`P(29 < X<36.5)`

And replacing we got:

`z= (29-34)/(2.5)= -2`

`z= (36.5-34)/(2.5)= 1`

For this case below 2 deviation from the mean we have 2.5% and above 1 deviation from the mean we got 16% and then the percentage between -2 and 1 deviation above the mean we got: (100-16-2.5)% = 81.5%