The data in the table were collected from randomly selected flights at airports

Question

The data in the table were collected from randomly selected flights at airports in three cities and indicate the number of minutes that each plane was behind schedule at its departure. Perform a one-way ANOVA using alpha = 0.05 to determine if there is a difference in the average lateness of flights from these three airports. Perform a multiple comparison test to determine which pairs are different using alpha =0.05. What are the correct hypothesis for the one-way ANOVA test? H_o: mu_1 = mu_2 = mu_3 H_1: Not all the means are equal. H_0: mu_1 notequalto mu_2 notequalto mu_3 H_1: mu_1 notequalto mu_2 mu_3 H_0: Not all the means are equal. H_1: mu_1 = mu_2 = mu_3 H_0: mu_1 = mu_2 = mu_3 H_1: mu_1 notequalto mu_2 notequalto mu_3 Complete the ANOVA summary table below. (Round to three decimal places as needed.)

Explanation / Answer

Here we have given three groups so we use one way anova.

One way ANova is used for testing more than two means.

Here we have to test the hypothesis that,

H0 : ALl means are equal.

H1 : Atleast one of the mean is differ than 0.

Assume alpha = level of significance = 0.05

Descriptive statistics of your k=3 independent treatments:

One-way ANOVA of your k=3 independent treatments:

The p-value corresponing to the F-statistic of one-way ANOVA is lower than 0.05, suggesting that the one or more treatments are significantly different.

We used here Scheffe's multiple comparison test. This test would likely identify which of the pairs of treatments are significantly differerent from each other.

We define a statistic named TT as the ratio of unsigned contrast mean to contrast standard error.

It can be show that for contrasts that are treatment pairs (i,j) with unit coefficients,

Ti,j = Qi,j / sqrt(2)

where Qi,jis the Q-statistic that was created for the Tukey HSD test. This T-statistic has interesting properties.

1 - F(T^2/ k-1, k-1, v)

This provides a formula which directly leads to the Scheffé p-value corresponding to an observed value of T.

where F() is the cumulative F distribution with its two degrees of freedom parameters k1 and . Note that k is the number of treatments and is the degrees of freedom of error that were established earlier.

The Scheffé p-value of the observed T-statistic Ti,j is shown below for all relevant pairs of treatments, along with color coded Scheffé inference (red for insignificant, green for significant) based on the p-value.

Scheffé results

We see that treatment A Vs B is significant and remaining two pairs are insignificant.

Treatment A B C Input Data 21.0
25.0
47.0
39.0
34.0 18.0
16.0
12.0
6.0
17.0 24.0
7.0
17.0
23.0
30.0

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