Question

63. Sharks Here are the lengths in feet of 44 great white sharks:

(a) Enter these data into your calculator and make a histogram. Then calculate one-variable statistics. Describe the shape, center, and spread of the distribution of shark lengths.
(b) Calculate the percent of observations that fall within one, two, and three standard deviations of the mean. How do these results compare with the
68–95–99.7 rule?
(c) Use your calculator to construct a Normal probability plot. Interpret this plot.
(d) Having inspected the data from several different perspectives, do you think these data are approximately Normal? Write a brief summary of your
assessment that combines your findings from (a) through (c).

Short answer

(a) Shape: approximately bell-shaped and symmetric.
Center: Mean is $15.5884$and median is $15.75$.
Spread: standard deviation is $2.5499$

(b) The one, two, and three standard deviations is $68.2\%,95.5\%,100\%$

(c) The pattern in the Normal probability plot is roughly linear and roughly normal.

(d) The data are approximately Normal.

Step-by-step solution

Part (a) Step 1: Given information

To describe the shape, center, and spread of the distribution of shark lengths. And to enter the data into the calculator and make a histogram. Then calculate one-variable statistics.

Part (a) Step 2: Explanation

Let, the sharks have lengths of 44 feet.
Below are some descriptive statistics:

Variable	N	Mean	Standard deviation	Minimum	$Q_1$	M	$Q_3$	Maximum
Shark length	44	15,586	2,550	9,40	13,525	15.75	17.40	22.80

And the histogram as:

Part (a) Step 3: Explanation

Shark lengths are nearly symmetric, peaking at $16$, and range from $9.40$feet to $22.8$ feet at the maximum.
As a result, Shape: approximately bell-shaped and symmetric.
Center: Mean is $15.5884$and median is $15.75$
Spread: standard deviation is $2.5499$.

Part (b) Step 1: Given information

To calculate the percent of observations that fall within one, two, and three standard deviations of the mean.

Part (b) Step 2: Explanation

According to the output:
Mean is $\overline{x}=15.586$
Sample standard deviation is $s=2.550$
$\overline{x}-s=13.036$
$\overline{x}+s=18.136$
The percent of interval as:

$(68.26\%)68.2\%$

$\overline{x}-2s=10.487$
$\overline{x}+2s=20.686$
Then the percent of interval as:
$(95.44\%)95.5\%$
$\overline{x}-3s=7.937$
$\overline{x}+3s=23.236$

Part (b) Step 3: Explanation

Then the percent of interval as:
$\text{(99.73\%)}100.0\%$
Approximately $68.2$percent of the data values fall inside one standard deviation, according to the $68-95-99.7$criterion.
Within two standard deviations of the mean, $95.5$percent of the data values fall, and within three standard deviations of the mean, $100$ percent of the data values fall.
As a result,
One standard deviation is $68.2\%$
Two standard deviations is $95.5\%$
Three standard deviations is $100\%$.

Part (c) Step 1: Given information

To use the calculator to construct a Normal probability plot. And to interpret the plot.

Part (c) Step 2: Explanation

A Normal probability plot from Minitab is shown below:

Except for one small shark and one enormous shark length, the plot is reasonably linear, indicating that the Normal distribution is reasonable.
As a result, a Normal probability plot is made.
The pattern in the Normal probability plot is roughly linear and roughly normal.

Part (d) Step 1: Given information

To write a brief summary of the assessment that combines the findings from (a) through (c).

Part (d) Step 2: Explanation

The results of parts (a), (b), and (c) show that shark lengths are roughly normal.
Because the normal probability plot was roughly normal and the histogram was approximately bell-shaped, it was roughly normal.
As a result, the observations are approximately Normal.