Chapter 2, Problem R2.8RE

To determine

Whether the distribution of the percent of state residents who are age 65 and older approximately Normal.

Answer to Problem R2.8RE

Yes, the distribution of the percent of state residents who are age 65 and older approximately Normal.

Explanation of Solution

Given information:

  

According to 68 − 95 − 99.7 rule:

68% of the data of a normal distribution lies with 1 standard deviation from the mean.

95% of the data of a normal distribution lies with 2 standard deviation from the mean.

99.7% of the data of a normal distribution lies with 1 standard deviation from the mean.

Then

The general Normal density graph is represented as:

  

Note that

The sample has 40 out of 51 observations within 1 standard deviation of the mean.

That means

The sample has 78.43% observations within 1 standard deviation.

Between

  $\overline{x}-s_x=13.255-1.668=11.587$

And

  $\overline{x}+s_x=13.255+1.668=14.923$

Also note that

The sample has 48 out of 51 observations within 2 standard deviation of the mean.

That means

The sample has 94.12% observations within 2 standard deviation.

Between

  $\overline{x}-2s_x=13.255-2\left(1.668\right)=9.919$

And

  $\overline{x}+2s_x=13.255+2\left(1.668\right)=16.591$

Also note that

The sample has 50 out of 51 observations within 3 standard deviation of the mean.

That means

The sample has 98.04% observations within 3 standard deviation.

Between

  $\overline{x}-3s_x=13.255-3\left(1.668\right)=8.251$

And

  $\overline{x}+3s_x=13.255+3\left(1.668\right)=18.259$

We have that

78.43% is close to 68%.

94.12% is close to 95%.

98.04% is very close to 99.7%.

This implies

The data reasonably follows 68 − 95 − 99.7 rule.

Thus,

The distribution is approximately Normal.