In Exercises 1–3, use the following recent annual salaries (in millions of dollars) for players on the N.Y. Knicks professional basketball team.

23.4 22.5 11.5 7.1 6.0 4.1 3.3 2.8 2.6 1.7 1.6 1.3 0.9 0.9 0.6 1.

NY Knicks Salaries

a. Find the mean x.

b. Find the median.

c. Find the standard deviation s.

d. Find the variance.

e. Convert the highest salary to a z score.

f. What level of measurement (nominal, ordinal, interval, ratio) describes this data set?

g. Are the salaries discrete data or continuous data?

a)

Let X denote the annual salaries of the players.

The mean is given as follows.

\(\begin{array}{c}\bar x = \frac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}\\ = \frac{{23.4 + 22.5 + 11.5 + 7.1 + .... + 0.6}}{{15}}\\ = 6.02\end{array}\)

Therefore, the mean is $6.02 million.

b)

The median is the middlemost observation when values are arranged in order.

Arranging the given values in ascending order, we get

0.6,0.9,0.9,1.3,1.6,1.7,2.6,2.8,3.3,4.1,6,7.1,11.5,22.5,23.4.

For odd counts of observations, the median is the\({\left( {\frac{{n + 1}}{2}} \right)^{th}}\)observation.

Therefore, the median is the 8^th observation when the count is 15.

Thus, the median salary is $2.8 million.

c)

The sample standard deviation for the sampled salaries is given as follows.

\[\begin{array}{c}s = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( {0.6 - 6.02} \right)}^2} + {{\left( {0.9 - 6.02} \right)}^2} + ... + {{\left( {23.4 - 6.02} \right)}^2}}}{{15 - 1}}} \\ = 7.47\end{array}\]

Therefore, the sample standard deviation of the salaries is $7.47 million.

d)

The variance is the square of the standard deviation, which is computed as follows.

\(\begin{array}{c}{s^2} = \;{\left( {7.47} \right)^2}\\ = 55.73\end{array}\).

Therefore, the variance of the salaries is $55.73 million.

e)

The highest value of X, i.e., the salaries, is $23.4 million.

From the above parts, we get

\(\begin{array}{l}\bar x = 6.02\\s = 7.47\\x = 23.4\end{array}\)

The z-score corresponding to the highest salary is given as follows.

\(\begin{array}{c}z = \frac{{x - \bar x}}{s}\\ = \frac{{23.4 - 6.02}}{{7.47}}\\ = 2.33\end{array}\)

Therefore, the z-score is 2.33.

f)

There are four levels of measurement: nominal, ordinal, interval, and ratio.

The nominal data can only be categorized, while the ordinal can be categorized and sorted. Moreover, interval data additionally holds the property that the differences between observations are measurable.The ratio data consists of all the properties along with a meaningful zero.

Thus, salaries are ratio-scaled data.

g)

Data is continuous if the variable can take any value in the given range (including the ones in the decimal form); otherwise, it is discrete.

The salaries are discrete data as they cannot take all values in a specific range