Family Heights. In Exercises 1–5, use the following heights (in.) The data are matched so that each column consists of heights from the same family.

1. a. Are the three samples independent or dependent? Why?

b. Find the mean, median, range, standard deviation, and variance of the heights of the sons.

c. What is the level of measurement of the sample data (nominal, ordinal, interval, ratio)?

d. Are the original unrounded heights discrete data or continuous data?

The data for heights of three members of thesame family is recorded.

a.

Certain groups of sampled observations are independent if they are not related to each other. In case the observations are related to one another, the samples become dependent. The paired or matched observations are also considered as dependent.

In this case, the heights of the members of the same family are observed.The three groups of observations from father, mother, and son are matched. Thus, the three samples are independent.

b.

For n observations,\({x_i}\)the formulae for different measures are shown below:

Mean:

\(\bar x = \frac{{\sum {{x_i}} }}{n}\)

Median:The middle observation is the median, where the central location obtained by the formula:

\(M = \left\{ \begin{array}{l}{\left( {\frac{n}{2}} \right)^{th}};\;\;\;\;\;\;\;\;\;\;\;\;\;\;n\;{\rm{is}}\;{\rm{odd}}\\\frac{{{{\left( {\frac{n}{2}} \right)}^{th}} + {{\left( {\frac{{n + 1}}{2}} \right)}^{th}}}}{2};\;\;\;\;n\;{\rm{is}}\;{\rm{even}}\end{array} \right.\)

Range:The range is the difference between the maximum and minimum values of the dataset.

Standard deviation:

\(s = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \)

Variance is\({s^2}\).

Compute the values for the heights of sons.

Mean:

\(\begin{aligned}{c}\bar x &= \frac{{71 + 64 + 71 + ... + 71}}{8}\\ &= 69.7\;{\rm{in}}\end{aligned}\)

Median:

Since n is even, the location of the middlemost observation is obtainedas shown below:

\(\begin{aligned}{c}M &= \frac{{{{\left( {\frac{8}{2}} \right)}^{th}} + {{\left( {\frac{8}{2} + 1} \right)}^{th}}}}{2}\\ &= \frac{{4th + 5th}}{2}\\ &= \frac{{71 + 71}}{2}\\ &= 71.0\;{\rm{in}}\end{aligned}\)

Range:

\(\begin{aligned}{c}Range &= {\rm{Max}} - {\rm{Min}}\\ &= 71.7 - 64\\ &= 7.7\;{\rm{in}}\end{aligned}\)

Standard deviation:

\(\begin{aligned}{c}s &= \sqrt {\frac{{{{\left( {71 - 69.71} \right)}^2} + {{\left( {64 - 69.71} \right)}^2} + ... + {{\left( {71 - 69.71} \right)}^2}}}{{8 - 1}}} \\ &= \sqrt {\frac{{1.66 + 32.63 + ... + 1.66}}{7}} \\ &= 2.57\;{\rm{in}}{\rm{.}}\\ &\approx {\rm{2}}{\rm{.6 in}}{\rm{.}}\end{aligned}\)

Variance:

\(\begin{aligned}{c}{s^2} &= {\left( {2.57} \right)^2}\\ &= 6.6\;{\rm{i}}{{\rm{n}}^2}\end{aligned}\).

c.

The four levels of measurements are listed below:

Nominal: The data can only be categorized.

Ordinal: The data can be categorized and ordered.

Interval: The data can be categorized, ordered, and have a meaningful interpretation of differences between any two values.

Ratio: The data consists of all three properties of interval and have a meaningful interpretation of 0.

Since the variable height has all four properties (categorization, rank, meaningful differences, and true meaning of zero measure), it has a ratio scale of measurement.

d.

The continuous form of data is the quantitative variable thatcan take any value within a range, including all decimal numbers. A discrete data can take certain values of a distinct set.

The height of the family members is a quantitative variable where heights can take any decimal value in a range of numbers. Thus, it is continuous data.