91影视

The data for three mini chambers are provided.

a.

Let x represents the values of cooler.

Let y represents the values of control.

Let z represents the values of warmer.

The mean values are computed as,

The sample mean of x is computed as,

\(\begin{array}{c}\bar x &=& \frac{{\sum x }}{{{n_x}}}\\ &=& \frac{{1.59 + 1.43 + 1.88 + ... + 1.76}}{{15}}\\ &=& 1.71\end{array}\)

Thus, the sample mean of x is 1.71.

The sample mean of y is computed as,

\(\begin{array}{c}\bar y &=& \frac{{\sum y }}{{{n_y}}}\\ &=& \frac{{1.92 + 2.0 + 2.19 + ... + 1.9}}{{11}}\\ &=& 1.75\end{array}\)

Thus, the sample mean of y is 1.75.

The sample mean of z is computed as,

\(\begin{array}{c}\bar z &=& \frac{{\sum z }}{{{n_z}}}\\ &=& \frac{{2.57 + 2.60 + 1.93 + ... + 1.91}}{{14}}\\ &=& 2.08\end{array}\)

Thus, the sample mean of z is 2.08.

It can be observed that the sample mean for Warmer is greater than both Cooler and Control.

b.

Referring to part a,

The sample mean of x is 1.71.

The sample mean of y is 1.75.

The sample mean of z is 2.08.

The standard deviation is computed as,

For x,

\(\begin{array}{c}{s_x} &=& \sqrt {\frac{{\sum {{{\left( {{x_i} - \bar x} \right)}^2}} }}{{{n_x} - 1}}} \\ &=& \sqrt {\frac{{{{\left( {1.59 - 1.71} \right)}^2} + {{\left( {1.43 - 1.71} \right)}^2} + ... + {{\left( {1.76 - 1.71} \right)}^2}}}{{15 - 1}}} \\ &=& 0.40\end{array}\)

For y,

\(\begin{array}{c}{s_y} &=& \sqrt {\frac{{\sum {{{\left( {{y_i} - \bar y} \right)}^2}} }}{{{n_y} - 1}}} \\ &=& \sqrt {\frac{{{{\left( {1.92 - 1.75} \right)}^2} + {{\left( {2.0 - 1.75} \right)}^2} + ... + {{\left( {1.9 - 1.75} \right)}^2}}}{{11 - 1}}} \\ &=& 0.53\end{array}\)

For z,

\(\begin{array}{c}{s_z} &=& \sqrt {\frac{{\sum {{{\left( {{z_i} - \bar z} \right)}^2}} }}{{{n_z} - 1}}} \\ &=& \sqrt {\frac{{{{\left( {2.57 - 2.08} \right)}^2} + {{\left( {2.6 - 2.08} \right)}^2} + ... + {{\left( {1.91 - 2.08} \right)}^2}}}{{14 - 1}}} \\ &=& 0.78\end{array}\)

It can be observed that the standard deviation of the chamber cooler is less than the standard deviation of the chamber warmer and control.

c.

The fourth spread is computed as,

\({f_s} = {\rm{Upper}}\,{\rm{fourth}} - {\rm{Lower}}\;{\rm{fourth}}\)

where the upper fourth is the median value of the largest half and the lower fourth is the median value of the smallest half.

The number of observations for the cooler chamber is 15 (odd).

The number of observations for the control chamber is 11 (odd).

The number of observations for the warmer chamber is 14 (even).

Arranging the data in ascending order,

For cooler,

\(\begin{array}{l}{n_{smallest}} &=& 7\\{n_{{\rm{largest}}}} &=& 7\end{array}\)

The lower fourth is computed as,

\(\begin{array}{c}\frac{{{n_{smallest}} + 1}}{2} &=& \frac{{7 + 1}}{2}\\ &=& {4^{th}}value\end{array}\)

The 4^th value of the smallest half is 1.43.

This implies that the lower fourth is 1.43.

The upper fourth is computed as,

\(\begin{array}{c}\frac{{{n_{{\rm{largest}}}} + 1}}{2} &=& \frac{{7 + 1}}{2}\\ &=& {4^{th}}value\end{array}\)

The 4^th value of the largest half is 1.9.

This implies that the upper fourth is 1.9.

The fourth spread for the chamber cooler is computed as,

\(\begin{array}{c}{f_s} &=& {\rm{Upper}}\,{\rm{fourth}} - {\rm{Lower}}\;{\rm{fourth}}\\ &=& 1.9 - 1.43\\ &=& 0.47\end{array}\)

Thus, the fourth spread of cooler chamber is 0.47.

For control,

\(\begin{array}{l}{n_{smallest}} &=& 5\\{n_{{\rm{largest}}}} &=& 5\end{array}\)

The lower fourth is computed as,

\(\begin{array}{c}\frac{{{n_{smallest}} + 1}}{2} &=& \frac{{5 + 1}}{2}\\ &=& {3^{rd}}value\end{array}\)

The 3^rd value of the smallest half is 1.52.

This implies that the lower fourth is 1.52.

The upper fourth is computed as,

\(\begin{array}{c}\frac{{{n_{{\rm{largest}}}} + 1}}{2} &=& \frac{{5 + 1}}{2}\\ &=& {3^{rd}}value\end{array}\)

The 3^rd value of the largest half is 2.03.

This implies that the upper fourth is 2.03.

The fourth spread for the chamber cooler is computed as,

\(\begin{array}{c}{f_s} &=& {\rm{Upper}}\,{\rm{fourth}} - {\rm{Lower}}\;{\rm{fourth}}\\ &=& 2.03 - 1.52\\ &=& 0.51\end{array}\)

Thus, the fourth spread of control chamber is 0.51.

For warmer,

\(\begin{array}{l}{n_{smallest}} &=& 7\\{n_{{\rm{largest}}}} &=& 7\end{array}\)

The lower fourth is computed as,

\(\begin{array}{c}\frac{{{n_{smallest}} + 1}}{2} &=& \frac{{7 + 1}}{2}\\ &=& {4^{th}}value\end{array}\)

The 4^th value of the smallest half is 1.91.

This implies that the lower fourth is 1.91.

The upper fourth is computed as,

\(\begin{array}{c}\frac{{{n_{{\rm{largest}}}} + 1}}{2} &=& \frac{{7 + 1}}{2}\\ &=& {4^{th}}value\end{array}\)

The 4^th value of the largest half is 2.6.

This implies that the upper fourth is 2.6.

The fourth spread for the chamber cooler is computed as,

\(\begin{array}{c}{f_s} &=& {\rm{Upper}}\,{\rm{fourth}} - {\rm{Lower}}\;{\rm{fourth}}\\ &=& 2.6 - 1.91\\ &=& 0.69\end{array}\)

Thus, the fourth spread of the warmer chamber is 0.69.

It can be observed that the fourth spread of the chamber cooler is less than the fourth spread of the chamber warmer and control.

Arranging the data in ascending order to compute the value of median,

For Cooler chamber,

For the odd number of observations, the median value is computed as,

\(\begin{array}{c}\tilde x &=& \;{\left( {\frac{{n + 1}}{2}} \right)^{th}}\;ordered\;value\\ &=& \;{\left( {\frac{{15 + 1}}{2}} \right)^{th}}\;ordered\;values\\ &=& \;{\left( 8 \right)^{th}}\;ordered\;values\\ &=& 1.76\end{array}\)

Thus, the median value is 1.76.

For Control chamber,

For the odd number of observations, the median value is computed as,

\(\begin{array}{c}\tilde y &=& \;{\left( {\frac{{n + 1}}{2}} \right)^{th}}\;ordered\;value\\ &=& \;{\left( {\frac{{11 + 1}}{2}} \right)^{th}}\;ordered\;values\\ &=& \;{\left( 6 \right)^{th}}\;ordered\;values\\ &=& 1.9\end{array}\)

Thus, the median value is 1.9.

For Warmer chamber,

For the even number of observations, the median value is computed as,

\(\begin{array}{c}\tilde z &=& average\;of\;{\left( {\frac{n}{2}} \right)^{th}}and\;{\left( {\frac{n}{2} + 1} \right)^{th}}\;ordered\;values\\ &=& average\;of\;{\left( {\frac{{14}}{2}} \right)^{th}}and\;{\left( {\frac{{14}}{2} + 1} \right)^{th}}\;ordered\;values\\ &=& average\;of\;{\left( 7 \right)^{th}}and\;{\left( 8 \right)^{th}}\;ordered\;values\\ &=& \frac{{2.3 + 2.31}}{2}\\ &=& 2.305\end{array}\)

Thus, the median value is 2.305.

Referring to part c,

For Cooler chamber,

The five number summary is,

The lower fourth is 1.43.

The upper fourth is 1.9.

The fourth spread is 0.47.

The smallest observation is 0.83

The largest observation is 2.41.

The median value is 1.76.

For Control chamber,

The five number summary is,

The lower fourth is 1.52.

The upper fourth is 2.03.

The fourth spread is 0.51.

The smallest observation is 0.53

The largest observation is 2.45.

The median value is 1.9.

For Warmer chamber,

The five number summary is,

The lower fourth is 1.91.

The upper fourth is 2.6.

The fourth spread is 0.69.

The smallest observation is 0.12.

The median value is 2.305.

The largest observation is 2.86.

Steps to construct a boxplot are as follows,

1)Compute the values of the five-number summary (smallest value, lower fourth, median, upper fourth, largest value).

2)Construct a line segment from the smallest value to the largest value of the dataset.

3) Construct a rectangular box from the lower fourth to upper fourth and draw a line in the box at the median value.

The boxplot is represented as,

The red boxplot represents the Cooler chamber, blue represents the Control chamber and green represents the Warmer chamber.

The features that can be observed from the above represented boxplot are,

1)There is no outlier present in the cooler chamber whereas there are outliers in other chambers.

2)The distribution of the cooler and control chamber is positively skewed whereas, for the warmer chamber, the distribution is approximately symmetric.

3)The average temperature for the warmer chamber is greater as compared to other two chambers.