Let’s say you randomly select 7 flights from 4 different airlines to examine if
ID: 3153704 • Letter: L
Question
Let’s say you randomly select 7 flights from 4 different airlines to examine if there are significant differences in on-time performance among the airlines. The dependent variable is the number of minutes that a flight is late (so a negative number means the flight is early, and a zero value means the flight was on-time). Here are the (hypothetical) data:
North-South Airline
Southern Skies
Central West
Happy Flier
1
25
15
-10
2
10
8
-8
1
15
5
-3
0
5
24
0
-10
-4
8
-24
2
0
10
-4
-3
14
8
-2
1. Use SPSS to produce an appropriate graph that shows the mean performance of each airline. Copy and paste the graph from the SPSS output below.
2. State the null and alternative hypotheses using symbols and/or words.
3. Using SPSS, determine whether there is a significant difference in on-time performance among the different airlines (alpha = .05), be sure to include the SPSS output for the ANOVA summary table.
4. State your conclusion and include a mean and SD table in APA style.
5. Which airlines (if any) have significant (alpha = .05) pair-wise differences in on-time performance? To fully answer this, you will need to conduct a Tukey’s HSD test as a post hoc test using SPSS (be sure to include the output below). Write the conclusions of this post hoc test in APA style. Additionally, provide clear statements in plain English about the significant differences in on-time performance among the airlines
North-South Airline
Southern Skies
Central West
Happy Flier
1
25
15
-10
2
10
8
-8
1
15
5
-3
0
5
24
0
-10
-4
8
-24
2
0
10
-4
-3
14
8
-2
Explanation / Answer
Your input data on k=4 independent treatments:
Descriptive statistics of your kk=4 independent treatments:
One-way ANOVA of your kk=4 independent treatments:
Conclusion from Anova:
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. The Tukey HSD test, Scheffé, Bonferroni and Holm multiple comparison tests follow. These post-hoc tests would likely identify which of the pairs of treatments are significantly differerent from each other.
Tukey HSD Test:
The p-value corrresponing to the F-statistic of one-way ANOVA is lower than 0.01 which strongly suggests that one or more pairs of treatments are significantly different. You have k=4 treatments, for which we shall apply Tukey's HSD test to each of the 6 pairs to pinpoint which of them exhibits statistially significant difference.
We first establish the critical value of the Tukey-Kramer HSD Q statistic based on the k=4 treatments and =24 degrees of freedom for the error term, for significance level = 0.01 and 0.05 (p-values) in the Studentized Range distribution.
We obtain these ctitical values for Q, for of 0.01 and 0.05, as
Qcritical=0.01,k=4,=24= 4.9071 and
Qcritical=0.05,k=4,=24 = 3.9015, respectively.
These critical values may be verified at several published tables of the inverse Studentized Range distribution, such as this table at Duke University.
Next, we establish a Tukey test statistic from our sample columns to compare with the appropriate critical value of the studentized range distribution.
We calculate a parameter for each pair of columns being compared, which we loosely call here as the Tukey-Kramer HSD Q-statistic, or simply the Tukey HSD Q-statistic, as:
Qi,j = | xibar - xjbar | / si,j
where the denominator in the above expression is:
si,j = ^/ sqrt(Hi,j) i,j=1,…,k;ij.
The quantity Hi,j is the harmonic mean of the number of observations in columns labeled i and j. Note that when the sample sizes in the columns are equal, then their harmonic mean is simply the common sample size. When the sample sizes of columns in a pair being compared are different, the harmonic mean lies somewhere in-between the two sample sizes. The relvant harmonic mean is required for applying the Tukey-Kramer procedure for columns with unequal sample sizes. The quantity ^ = 7.4769 is the square root of the Mean Square Error = 55.9048 determined in the precursor one-way ANOVA procedure. Note that ^ is same across all pairs being compared.
The only factor that varies across pairs in the computation of si,j=^*sqrt(Hi,j) is the denominator, which is the harmonic mean of the sample sizes being compared.
The test of whether the NIST Tukey-Kramer confidence interval includes zero is equivalent to evaluating whether
Qi,j > Qcritical the latter determined according to the desired level of significance (p-value), the number of treatments k and the degrees of freedom for error , as described above.
post-hoc Tukey HSD Test results :
k=4 treatments
degrees of freedom for the error term =24
Critical values of the Studentized Range Q statistic:
Qcritical=0.01,k=4,=24= 4.9071 and
Qcritical=0.05,k=4,=24 = 3.9015, respectively.
We present below color coded results (red for insignificant, green for significant) of evaluating whether
Qi,j > Qcritical for all relevant pairs of treatments. In addition, we also present the significance (p-value) of the observed Q-statistic Qi,j.
Tukey HSD results
Treatment A B C D Input Data 1.02.0
1.0
0.0
-10.0
2.0
-3.0 25.0
10.0
15.0
5.0
-4.0
0.0
14.0 15.0
8.0
5.0
24.0
8.0
10.0
8.0 -10.0
-8.0
-3.0
0.0
-24.0
-4.0
-2.0
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