Question

the results of a certain medical test are normally distributed with a mean of 124 and a standard deviation of 11. convert the given results into z - scores, and then use the accompanying table of z - scores and percentiles to find the percentage of people with readings between 113 and 135.
click the icon to view the table of z - scores and percentiles.
the percentage of people with readings between 113 and 135 is
(round to two decimal places as needed.)

Explanation:

Step1: Calculate the z - score for \(x = 113\)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 124\) (mean) and \(\sigma=11\) (standard deviation).
For \(x = 113\), \(z_1=\frac{113 - 124}{11}=\frac{- 11}{11}=-1\)

Step2: Calculate the z - score for \(x = 135\)

Using the same formula \(z=\frac{x-\mu}{\sigma}\)
For \(x = 135\), \(z_2=\frac{135 - 124}{11}=\frac{11}{11}=1\)

Step3: Find the percentage using the z - score table

The percentage of data to the left of \(z=-1\) is \(15.87\%\) (from the standard normal table).
The percentage of data to the left of \(z = 1\) is \(84.13\%\) (from the standard normal table).
The percentage of data between \(z=-1\) and \(z = 1\) is \(P(-1<Z<1)=P(Z < 1)-P(Z<-1)\)
\(P(-1<Z<1)=84.13\% - 15.87\%=68.26\%\)

Answer:

\(68.26\)