Q.32 - Elementary Statistics

Question

 High and Low Temperatures. The U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. [Note: Do parts (a)-(c) for both the maximum and minimum temperatures.]

a. Obtain the mean, median, and mode(s) of the data. Determine which of these measures of center is best, and explain your answer:

b. Determine the range and sample standard deviation of the data.

c. Find the five-number summary and interquartile range of the data.

d. Identify potential outliers. if any:

e. Obtain and interpret a boxplot.

Step-by-Step Solution

Answer

(a) Maximum temperature

Mean $65.3268$, Median$=64.4$,

Mode$=54.5,55.9,57.6,59.8,62.6,63.6,65.1,67.4,67.8,70.6$

   Minimum temperature

Mean$=45.5197$ Median$=44.2$$=44.2$

Mode$=35.8,39.8,41.2,43.2,44.8,47.4,48.6,52.5$

(b) Maximum temperature 

 Standard deviation$=8.7228$

Minimum temperature

 Standard deviation$=9.3426$

(c) Interquartile range (Maximum)$=12.7$

    Interquartile range (minimum) $=11.6$

(d) There are no outliers in Maximum temperature

     $69.1,70.2\&74.2$are the outliers.

(e)Maximum temperature 

Minimum temperature

Part (a) Step 1: Given Information

Given in the question that the U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. we  have to  obtain the mean, median, and mode(s) of the data .

Part (a) Step 2: Explanation

Maximum temperature 

Mean

$$\begin{gathered} \overline{x}=\frac{47.6+48.7+49.6+\dots +84.5+84.7+85.5}{71} \\ =\frac{4638.2}{71} \\ \approx 65.3268 \end{gathered}$$

Median is defined as the central value. so the median is $64$

Mode is defined as the number ,which is coming maximum number of times.

so, mode$=54.5,55.9,57.6,59.8,62.6,63.6,65.1,67.4,67.8,70.6$

Minimum temperature 

 

Mean

$$\begin{gathered} \overline{x}=\frac{29.3+30.1+30.5+\dots +69.1+70.2+74.2}{71} \\ =\frac{3231.9}{71} \\ \approx 45.5197 \end{gathered}$$

Median $=44.2$

Mode$=35.8,39.8,41.2,43.2,44.8,47.4,48.6,52.5$

Part (b) Step 1: Given Information

Given in the question that the U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. we  have to determine the range and sample standard deviation of the data. 

Part (b) Step 2: Explanation

Maximum temperature  

The standard deviation can be computed as 

$$\begin{gathered} s=\sqrt{\frac{(47.6-65.3369{)}^2+\dots +(85.5-65.3268{)}^2}{71-1}} \\ \approx 8.7228 \end{gathered}$$

Minimum temperature 

The standard deviation can be computed as 

$$\begin{gathered} s=\sqrt{\frac{(29,3-45.5197{)}^2+\dots +(74.2-45.5197{)}^2}{71-1}} \\ \approx 9.3426 \end{gathered}$$

Part (c) Step 1: Given Information

Given in the question that the U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. we  have to find the five-number summary and interquartile range of the data .

Part (c) Step 2: Explanation

Maximum temperature 

Minimum value$=47.6$

Quartile$1=58.4$

Median$=64$

Quartile $3=71.1$

Maximum value$=85.5$

Interquartile range is

                          $$\begin{gathered} =Q_3-Q_1 \\ =71.1-58.4 \\ =12.7 \end{gathered}$$ 

Minimum temperature 

Minimum value$=29.3$

Quartile $1=39.8$

Median$=44.2$

Quartile$3=51.4$

Maximum value$=74.2$

Interquartile range

                                $$\begin{gathered} =Q_3-Q_1 \\ =51.4-39.8 \\ =11.6 \end{gathered}$$                

Part (d) Step 1: Given Information

Given in the question that the U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. we  have to identify potential outliers. 

Part (d) Step 2: Explanation

Maximum temperature 

$Q_3+1.5\left(IQR\right)=71.1+1.5(12.7)=90.15$

$Q_1-1.5\left(IQR\right)=58.4-1.5(12.7)=39.35$

Minimum temperature 

$Q_3+1.5\left(IQR\right)=51.4+1.5(11.6)=68.8$

$Q_1-1.5\left(IQR\right)=51.4-1.5(11.6)=22.4$

Part (e) Step 1: Given Information

Given in the question that the U.S. National Oceanic and Atmospheric Administration publishes temperature data in Climatography of the United States. According to that document, the annual average maximum and minimum temperatures for selected cities in the United States are as provided on the WeissStats site. we  have to obtain and interpret a boxplot. 

Part (e) Step 2: Explanation

Maximum temperature  

Minimum temperature