In Exercises 1–5, refer to the following list of departure delay times (min) of American Airline flights from JFK airport in New York to LAX airport in Los Angeles. Assume that the data are samples randomly selected from larger populations.

Flight 3	22	-11	7	0	-5	3	-8	8
Flight 19	19	-4	-5	-1	-4	73	0	1
Flight 21	18	60	142	-1	-11	-1	47	13

Exploring the Data Include appropriate units in all answers.

a. Find the mean for each of the three flights.

b. Find the standard deviation for each of the three flights.

c. Find the variance for each of the three flights.

d. Are there any obvious outliers?

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

The delay times for three flights is recorded below.

a.

The formula to compute the mean of any set of observations is:

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

Using the observations, the mean for flights 3,19 and 21 are computed as follows,

\(\begin{array}{c}{{\bar x}_1} = \frac{{22 + \left( { - 11} \right) + 7 + ... + 8}}{8}\\ = 2.0\;\min \end{array}\)

\(\begin{array}{c}{{\bar x}_2} = \frac{{19 + \left( { - 4} \right) + \left( { - 5} \right) + ... + 1}}{8}\\ = 9.9\;\min \\{{\bar x}_3} = \frac{{18 + 60 + 142 + ... + 13}}{8}\\ = 33.4\;\min \end{array}\)

Thus the mean of flight 3 is 2.0 min, flight 19 is 9.9 min, and flight 21 is 33.4 min.

b.

The formula for the standard deviation of sample values are:

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

Using the observations, the standard deviation for flights 3,19 and 21 are computed as follows,

\(\begin{array}{c}{s_1} = \sqrt {\frac{{{{\left( {22 - 2.0} \right)}^2} + {{\left( { - 11 - 2.0} \right)}^2} + ... + {{\left( {8 - 2.0} \right)}^2}}}{{8 - 1}}} \\ = 10.6\,\min \\{s_2} = \sqrt {\frac{{{{\left( {19 - 9.9} \right)}^2} + {{\left( { - 4 - 9.9} \right)}^2} + ... + {{\left( {1 - 9.9} \right)}^2}}}{{8 - 1}}} \\ = 26.6\,\min \end{array}\)

\(\begin{array}{c}{s_3} = \sqrt {\frac{{{{\left( {18 - 33.4} \right)}^2} + {{\left( {60 - 33.4} \right)}^2} + ... + {{\left( {13 - 33.4} \right)}^2}}}{{8 - 1}}} \\ = 50.3\,\min \end{array}\)

Thus the standard deviation of flight 3 is 10.6 min, flight 19 is 26.6minand flight 21 is 50.3min.

c.

The formula for the variance of sample values is the square of the standard deviation measure.

Using the observations, the standard deviation for flights 3,19 and 21 are computed as follows,

\(\begin{array}{c}s_1^2 = {10.6^2}\,\\ = 112.0{\min ^2}\\s_2^2 = {26.6^2}\\ = 709.8\,{\min ^2}\end{array}\)

\(\begin{array}{c}s_3^2 = {50.3^2}\\ = 2525.4\,{\min ^2}\end{array}\)

Thus the variance of flight 3 is 112.0 min square, flight 19 is 709.8 min square, and flight 21 is 2525.4 min square.

d.

An outlier is the set of extreme values compared to other set of values in the dataset.

Visually, the most extreme value in the dataset is 142 min, which is extremely large compared to other delay times values.

e.

There are four levels of measurement:

• Nominal: the data that can only be categorized.
• Ordinal: the data which can be categorized and arranged in a particular order.
• Interval: the data which can be categorized, ordered, and have well-defined differences between observations.
• Ratio: the data with the characteristic that it can be categorized and ordered has well-defined differences between observations and true meaning of zero value.

The observations of departure delay times are measured on a ratio scale as they can be categorized, ordered, and have defined gaps between observations.

The true value of 0 min for departure delay time implies that in the flight, there is no delay in the departure.