Question

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: symmetric and roughly bell-shaped.

Center: Mean is $15.5884$and median is $15.75$

Spread: standard deviation is $2.5499$.

b). One standard deviations $=68.2\%$.

Two standard deviations $=95.5\%$.

Threestandard deviations $=100\%$.

c). Except for one small and one huge shark length, the figure is reasonably linear, demonstrating that the Normal distribution is reasonable.

d). The histogram was roughly bell-shaped, and the normal probability plot was roughly normal.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

The following are descriptive statistics and a histogram:

Shark lengths are nearly symmetric, peaking at $16$, and range from $9.40$feet to $22.8$feet.

Shape: symmetric and roughly bell-shaped.

Center: Mean is $15.5884$ and median is $15.75$.

Spread: standard deviation is $2.5499$.

Part (b) Step 1: Given Information

Given data:

Part (b) Step 2: Explanation

Mean, $\overline{x}=15.586$

Sample standard deviation, $s=2.550$

$\overline{x}-s=13.036$

$\overline{x}+s=18.136$

Percent of interval

$(68.26\%)68.2\%$

$\overline{x}-2s=10.487$

$\overline{x}+2s=20.686$

Percent of interval

$(95.44\%)95.5\%$

$\overline{x}-3s=7.937$

$\overline{x}+3s=23.236$

Percent of interval

$(99.73\%)100.0\%$

According to the $68-95-99.7$rule, approximately $68.2\%$of the data values fall within one standard deviation. $95.5\%$of the data values fall within two standard deviations, and $100\%$ of the data values fall within three standard deviations of the mean.

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Explanation

The following is a Minitab normal probability plot:

Except for one small and one huge shark length, the figure is reasonably linear, demonstrating that the Normal distribution is reasonable.

Part (d) Step 1: Given Information

Given data:

Part (d) Step 2: Explanation

The results of parts (a), (b), and (c) show that shark lengths are about average.

The histogram was roughly bell-shaped, and the normal probability plot was roughly normal.