Question

A normal curve with a mean of 500 and a standard deviation of 100 is shown. Shade the region under the curve within one standard deviation of the mean. How much of the data falls within the shaded region?

Answer 1

And for this case we want to find how much of the data within the shaded region given in the figure attached.

And we can use the zscore formula given by:

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

And if we find the z score for the limits we got:

`z = (400-500)/(100)=-1`

`z = (600-500)/(100)= 1`

And we want to find this probability using the normal standard distribution or excel:

`P(-1<z<1) = P(z<1) -P(z<-1) =0.841-0.159 = 0.682`

So we would expect about 68.2 % of the data within one deviation from the mean

Explanation:

For this case we know that the variable of interest let's say x follows a normal distribution with the following parameters:

`X \sim N(\mu = 500, \sigma =100)`

And for this case we want to find how much of the data within the shaded region given in the figure attached.

And we can use the zscore formula given by:

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

And if we find the z score for the limits we got:

`z = (400-500)/(100)=-1`

`z = (600-500)/(100)= 1`

And we want to find this probability using the normal standard distribution or excel:

`P(-1<z<1) = P(z<1) -P(z<-1) =0.841-0.159 = 0.682`

So we would expect about 68.2 % of the data within one deviation from the mean