Q. 3.209 - Elementary Statistics

Question

Atlantic Basin Hurricanes. The Tropical Cyclone Report, a publication of the National Hurricane Center, contains comprehensive information on each tropical cyclone, including synoptic history, meteorological statistics, casualties, and damages. A hurricane is a tropical cyclone with winds that have reached a constant speed of $74 miles per hour$ or more. During one year, there were $10$ Atlantic basin hurricanes. Their maximum wind speeds, in miles per hour ($mph$), were as shown in the following table.

Consider these storms a population of interest. Obtain the following parameters for the maximum wind speeds. Use the appropriate mathematical notation for the parameters to express your answers.

a. Mean

b. Standard deviation

c. Median

d. Mode

e. IQR

Step-by-Step Solution

Answer

(a) The Mean is, $\mu =97$

(b) The Standard deviation is, $\sigma =13.0767$

(c) The median is, $\eta =95$

(d) The mode is, 

           Mode$=80,90,115$

(e) The $IQR=25$

Part (a) Step 1: Given informnation

We are given that, 

The data set for maximum wind speed is, 
$x_i=85, 80, 100, 115, 110, 90, 80, 90, 105, 115$ 

No. of values are,

   $N=10$

Part (a) Step 2: Simplify

We know that,

The mean is given as,

$$\begin{gathered} \mu =\frac{{\displaystyle \sum _{}}{x}_i}{N} \\ =\frac{85+100+110+80+105+80+115+90+115}{10} \\ =\frac{970}{10}=97 \end{gathered}$$

Part (b) Step 1: Given information

We need to find out the standard deviation.

Part (b) Step 2: Simplify

We know that the standard deviation is the square root of variance which is given as,

xi	Deviation	Squared deviation
$85$	$12$	$144$
$100$	$3$	$9$
$110$	$13$	$169$
$80$	$17$	$289$
$105$	$8$	$64$
$80$	$17$	$289$
$115$	$18$	$324$
$90$	$7,7$	$49,49$
$115$	$18$	$324$

Now, the sum of squared deviations is,   $\sum _{}^{}{\left(x_i-\mu \right)}^2=144+9+169+289+64+289+324+49+49+324=1710$

Therefore, the Standard deviation is,

      $\sigma =\sqrt{\frac{\sum _{}^{}{\left(x_i-\mu \right)}^2}{N}}=\sqrt{\frac{1710}{10}}=\sqrt{171}\approx 13.0767$

Part (c) Step 1: Given information

We need to find out the median of the data

Part (c) Step 2: Simplify

Data in the ordered list is,

  $80, 80, 85, 90, 90, 100, 105, 110, 115, 115$

The median of the sorted list is the middle value of the list,

Therefore, median is, 

   $\eta =\frac{90+100}{2}=95$

Part(d) Step 1: Given information

We need to find out the mode of given data.

Part(d) Step 2: Simplify

We know that mode is the most occurring value in the data set.

Therefore, mode is, $80, 90, 115$

Part (e) Step 1: Given information

We need to find the value of IQR

Part (e) Step 2: Simplify

As we know, 

The first quartile is the median of data below the median,

Therefore, $Q_1=85$

And the third quartile is the median of data above median,

Therefore, $Q_3=110$

Hence, the interquartile range is, 

     $IQR=Q_3-Q_1=110-85=25$