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Magazine Chapter 13: Q2CRE (page 597)
Test for Normality Use the departure delay times for Flight 19 and test for normality using a normal quantile plot.
Short Answer
The following normal quantile plot is constructed:
As the majority of points do not lie along a straight line, the departure delay times of Flight 19 do not come from a normally distributed population.
Step by step solution
Given information
The departure delay times (in minutes) for Flight 19 are given.
Normal quantile plot
Follow the given steps to construct a normal quantile plot:
Arrange the given values in ascending order as shown:
-5 | -4 | -4 | -1 | 0 | 1 | 19 | 73 |
Compute the cumulative areas to the left for each sample as follows:
Sample value | Areas to the left |
-5 | \(\begin{array}{c}\frac{1}{{2n}} = \frac{1}{{2\left( 8 \right)}}\\ = 0.0625\end{array}\) |
-4 | \(\begin{array}{c}\frac{3}{{2n}} = \frac{3}{{2\left( 8 \right)}}\\ = 0.1875\end{array}\) |
-4 | \(\begin{array}{c}\frac{5}{{2n}} = \frac{5}{{2\left( 8 \right)}}\\ = 0.3125\end{array}\) |
-1 | \(\begin{array}{c}\frac{7}{{2n}} = \frac{7}{{2\left( 8 \right)}}\\ = 0.4375\end{array}\) |
0 | \(\begin{array}{c}\frac{9}{{2n}} = \frac{9}{{2\left( 8 \right)}}\\ = 0.5625\end{array}\) |
1 | \(\begin{array}{c}\frac{{11}}{{2n}} = \frac{{11}}{{2\left( 8 \right)}}\\ = 0.6875\end{array}\) |
19 | \(\begin{array}{c}\frac{{13}}{{2n}} = \frac{{13}}{{2\left( 8 \right)}}\\ = 0.8125\end{array}\) |
73 | \(\begin{array}{c}\frac{{15}}{{2n}} = \frac{{15}}{{2\left( 8 \right)}}\\ = 0.9375\end{array}\) |
The corresponding z-scores of the areas computed above are tabulated below:
Areas | z-scores |
0.0625 | -1.534 |
0.1875 | -0.887 |
0.3125 | -0.489 |
0.4375 | -0.157 |
0.5625 | 0.157 |
0.6875 | 0.489 |
0.8125 | 0.887 |
0.9375 | 1.534 |
Now, plot the original data values on the x-axis and the corresponding z-scores on the y-axis.
Sample values (x) | z-scores (y) |
-5 | -1.534 |
-4 | -0.887 |
-4 | -0.489 |
-1 | -0.157 |
0 | 0.157 |
1 | 0.489 |
19 | 0.887 |
73 | 1.534 |
- Mark the values -20, 0, 20, 40, 鈥︹��., 80 on the horizontal scale. Label the axis as 鈥淪ample Values鈥�.
- Mark the values -2.000, -1.500, -1.000, 鈥︹��.., 2.000 on the vertical axis. Label the axis as 鈥渮-score鈥�.
- Place a dot for the values of the z-scores corresponding to the sample values on the x-axis.
- Draw a straight trend line.
The following normal quantile plot is obtained:
Since most of the points do not lie along a straight line, the departure delay times of Flight 19 do not come from a normally distributed population.
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