Question

Two marathon training procedures are tried for comparison purposes. Their efficacy is to be determined in a marathon race. Assume race times are given in minutes. Use α=0.056α=0.056 for all calculations.
Procedure_A   Procedure_B
146   144
147   147
149   150
142   147
144   141
147   147
143   147
148   148
147   148
146   151
147   145
144   141
144   153
147   150
140   141
151
144
156
150
153
155
140
   138
146
141
151

(a) Do the populations appear to be normal? Hint: use a probability plot to decide
A. Yes, procedure B appears to be normal but procedure A does not appear to be normal
B. Yes, Both procedure A and procedure B appear to be normal
C. Yes, procedure A appears to be normal but procedure B does not appear to be normal
D. No, Both procedure A and procedure B appear to be normal
E. No, neither procedure A appears to be normal nor procedure B appears to be normal
F. Yes, Both procedure A and procedure B appear to be non-normal


(b) Choose the most appropriate statistical hypotheses to see which procedure results in longer marathon times (be very careful in how you formulate your hypothesis, this one is tricky! Use the sample data to make any assumptions necessary.)
A. H0:μ1=μ2HA:μ1>μ2H0:μ1=μ2HA:μ1>μ2
B. H0:μ1=μ2HA:μ1<μ2H0:μ1=μ2HA:μ1<μ2
C. H0:μD=0,HA:μD<0H0:μD=0,HA:μD<0
D. H0:μ1=μ2HA:μ1≠μ2H0:μ1=μ2HA:μ1≠μ2
E. H0:μD=0,HA:μD≠0H0:μD=0,HA:μD≠0
F. H0:μD=0,HA:μD>0H0:μD=0,HA:μD>0


(c) Using technology available to you, test to see if the variances are equal or not?
A. There appears to be more variation in the 2nd procedure then in the 1st.
B. They appear to be equal.
C. There appears to be an unequal amount of variation in the 2nd procedure then in the 1st.
D. There appears to be more variation in the 1st procedure then in the 2nd.


(d) Report the p-value of the test you ran in (c) use at least three decimals in your answer (this is from Levene's test).
P-value =

(e) Test the statistical hypotheses in (b) by carrying out the appropriate statistical test. Find the value of the test statistic for this test, use at least two decimals in your answer (from your test of middle).
Test Statistic =

(f) Determine the PP-value for this test, to at least three decimal places (from your test of middle).

Answer 1

(a)

From above probability plots, we see that both data sets are obtained from two independent normal populations.

Hence Option B. Yes, Both procedure A and procedure B appear to be normal.

(b)

Option: D. H0:μ1=μ2 HA:μ1≠μ2

(c)

Test and CI for Two Variances: Procedure_A, Procedure_B

Method

Null hypothesis Sigma(Procedure_A) / Sigma(Procedure_B) = 1
Alternative hypothesis Sigma(Procedure_A) / Sigma(Procedure_B) not = 1
Significance level Alpha = 0.05

Statistics

Variable N StDev Variance
Procedure_A 26 4.526 20.486
Procedure_B 15 3.697 13.667

Ratio of standard deviations = 1.224
Ratio of variances = 1.499

Tests

Test
Method DF1 DF2 Statistic P-Value
F Test (normal) 25 14 1.50 0.433
Levene's Test (any continuous) 1 39 0.70 0.409

Since P-value>0.05 so

Option: B. They appear to be equal.

(d) P-value=0.409

(e)

Two-Sample T-Test and CI: Procedure_A, Procedure_B

Two-sample T for Procedure_A vs Procedure_B

N Mean StDev SE Mean
Procedure_A 26 146.38 4.53 0.89
Procedure_B 15 146.67 3.70 0.95

Difference = mu (Procedure_A) - mu (Procedure_B)
Estimate for difference: -0.28
95% CI for difference: (-3.07, 2.50)
T-Test of difference = 0 (vs not =): T-Value = -0.20 P-Value = 0.839 DF = 39
Both use Pooled StDev = 4.2471

Test Statistic = -0.20

(f) P-value for this test=0.839