91影视

The data are provided that consists of 35 observations, so the sample size is 35.

The five-summary are smallest\({x_i}\), lower fourth, median, upper fourth and largest\({x_i}\).

Since sample size is odd, the median is the\({\left( {\frac{n}{2} + 1} \right)^{th}}\)ordered value when the data are in ascending order as below.

The smallest value is: 15.3 and the largest value is: 23.78.

Let\(\tilde x\)be the required median. Use the formula to calculate median,

\(\begin{aligned}\tilde x &= {\left( {\frac{{n + 1}}{2}} \right)^{th}}\,ordered\,value\\ &= {\left( {\frac{{35 + 1}}{2}} \right)^{th}}\\ &= {18^{th\,}}observation\\ &= 19.2\end{aligned}\)

The lower fourth is the median of smallest half of the data as the median of the data is\(\tilde x = 19.2\)so each half contains 18 values.

Since\(n = 18\)is even, calculate the median using the formula:

\(\begin{aligned}\tilde x &= \frac{{{{\left( {\frac{n}{2}} \right)}^{th}} + {{\left( {\frac{n}{2} + 1} \right)}^{th}}\,ordered\,value}}{2}\\ &= \frac{{{{\left( {\frac{{18}}{2}} \right)}^{th}} + {{\left( {\frac{{18}}{2} + 1} \right)}^{th}}ordered\,value}}{2}\\ &= \frac{{\left( {18 + 18.68} \right)}}{2}\\ &= 18.34\end{aligned}\)

Similarly, the upper fourth is the median of largest half of the data as the median of the data is\(\tilde x = 19.2\)so it contains 18 values.

Since\(n = 18\)is even, calculate the median using the formula:

\(\begin{aligned}\tilde x &= \frac{{{{\left( {\frac{n}{2}} \right)}^{th}} + {{\left( {\frac{n}{2} + 1} \right)}^{th}}\,ordered\,value}}{2}\\ &= \frac{{{{\left( {\frac{{18}}{2}} \right)}^{th}} + {{\left( {\frac{{18}}{2} + 1} \right)}^{th}}ordered\,value}}{2}\\ &= \frac{{\left( {19.62 + 19.90} \right)}}{2}\\ &= 19.76\end{aligned}\)

Thus, the five-number summary to construct boxplot are as follows:

smallest\({x_i}\): 15.30, lower fourth: 18.34, median: 19.20, upper fourth: 19.76,

largest \({x_i}\): 23.78.

Following are the steps to construct a box-plot for the given data:

1. Open Minitab and enter the given data into the worksheet.
2. Choose Graph and select 鈥淏ox-plot鈥�.
3. Click on 鈥淪imple鈥� from the list of One Y and click 鈥淥k鈥�.
4. Double click on C1 observations to specify it in the graph variables box and click 鈥淥k鈥�.

From the boxplot, one can see that the distribution is slightly negatively skewed. Since the variability is quite large as there are two outliers on the upper end i.e. 23.25 and 23.78.