Question

Refrigerators Consumer Reports magazine collected data on the usable

capacity (in cubic feet) of a sample of $36$ side-by-side refrigerators. Here are the data:

A histogram of the data and summary statistics are shown here. Is this distribution of refrigerator capacities approximately Normal? Justify your answer based on the graph and the $68–95–99.7$ rule.

Short answer

The distribution of refrigerator capacities is approximately normal.

Step-by-step solution

Given information

Concept

Use of $68-95-99.7$ rule.

Calculation

According to $68-95-99.7$rule:

$68$ percent of the data in a normal distribution is within $1$ standard deviation of the mean.

In a normal distribution, $95\%$of the data lies within $2$standard deviations of the mean.

A normal distribution has $99.7\%$of its data inside $1$ standard deviation of the mean.

Then

The general Normal density graph is represented as:

Calculation

Note that

$24$ of the $36$ observations in the sample are within one standard deviation of the mean.

That means

The sample has $66.67\%$ observations within $1$ standard deviation.

Between

$\overline{x}-s_x=15.825-1.217=14.608$

And

$\overline{x}+s_x=15.825+1.217=17.042$

Also, note that

$34$ of the $36$ observations in the sample are within $2$ standard deviations of the mean.

That means

The sample has $94.44\%$ observations within $2$ standard deviations.

Between

$\overline{x}-2s_x=15.825-2(1.217)=13.391$

And

$\overline{x}+2s_x=15.825+2(1.217)=18.259$

Also, note that

$36$ of the $36$ observations in the sample are within $3$ standard deviations of the mean.

That means

The sample has $100\%$ observations within $3$ standard deviations.

Between

$\overline{x}-3s_x=15.825-3(1.217)=12.174$

And

$\overline{x}-3s_x=15.825+3(1.217)=19.476$

We have that

$66.67\%$ is very close to $68\%$

$94.44\%$ is very close to $95\%$

$100\%$ is very close to $99.7\%$

This implies

The data reasonably follows $68-95-99.7$ rule.

Thus,

The distribution is approximately normal.