Chapter 12.4, Problem 61PPSE

Solution

To find: To determine the statements that are not true.

A.I and II are NOT true.

Given information:

Given,a set of data is normally distributed witha mean of 44and a standard deviation of 3.2.

Given statements are:

1. 68% of the values are between 37.6 and 50.4
2. 13.5% of the values are less than 40.8
3. 5% of the values are lower than 37.6 or higher than 50.4

Calculation:

One standard deviation below and above the mean are:

  $\begin{array}{l}44-3.2=40.8\\ 44+3.2=47.2\end{array}$

Two standard deviation below and above the mean are:

  $\begin{array}{l}44-2\left(3.2\right)=37.6\\ 44+2\left(3.2\right)=50.4\end{array}$

In a normal distribution, the data is distributed equally i.e., 50% on either side of the mean.

The data within the first standard deviation contains 34% of data on each side.

The data between the first standard deviation and second standard deviation contains 13.5% of data on each side.

The data between the second standard deviation and third standard deviation contains 2.35% of data on each side.

Here, 37.6 and 50.4are the second standard deviationson either sides of themean.

Hence the percent of data between 37.6 and 50.4is given by:

  $\begin{array}{l}p\approx 13.5+34+34+13.5\\ p\approx 95\%\end{array}$

Hence the statement I is NOT true.

Also, 40.8 is the first standard deviation on the left of the mean. The data to the right of this is $34+50=84\%$

So the data percent of values less than 40.8 is $100-84=16\%$

Hence the statement II is NOT true.

The percent of data between 37.6 and 50.4 is 95%. Hence the percent of data lower than 37.6 and higher than 50.4 is $100-95=5\%$

Hence the statement III is true.

Conclusion:

A.I and II are NOT true.