Question

Your boss tells you to analyze the comparative effectiveness of two existing UAVs at detecting particular “events” on the ground in the CENTCOM area of responsibility (AOR). To do so, you are given data from a series of experiments during which UAV “A” and UAV “B” were tested under controlled conditions. For each experiment, both UAVs were flown. The data is shown below. Target UAV “A” Score UAV “B” Score 1 72 93 2 77 88 3 42 36 4 18 29 5 11 10 Assume that the data are paired (i.e. tested against the same targets). At a significance level of? = 0.05, test the hypothesis that the two types of UAVs perform the same versus the alternative hypothesis that they are different. Hint: think paired data! Conduct a formal hypothesis test and report your conclusions.

Target	UAV “A” Score	UAV “B” Score
1	72	93
2	77	88
3	42	36
4	18	29
5	11	10

A) Parameter of interest: From the problem context, identify the parameter of interest.

B) Null hypothesis, H0: State the null hypothesis, H0 in terms of the parameter of interest

C) Alternative hypothesis, H1: Specify an appropriate alternative hypothesis, H1.

D) Test Statistic: Determine an appropriate test statistic (equation; state degrees if freedom if necessary).

E) Reject H0 if: State the rejection criteria for the null hypothesis for the given level of α.

F) Computations: Compute any necessary sample quantities, substitute these into the equations for the test statistic, and compute that value. Perform P-Value calculations.

G) Draw conclusions: Decide whether or not H0 should be rejected and report that in the problem context. Make a “real-world” statement about the outcome of the test (cannot just say “reject the null hypothesis”)

Answer 1

Solution:-

A) The parameter of interest is UAV score.

B)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d = 0

C)

Alternative hypothesis: u_d ≠ 0

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

Target	UAV "A" Score	UAV "B" Score	d	d - dbar	(d - dbar)^2
1	72	93	-21	-13.8	190.44
2	77	88	-11	-3.8	14.44
3	42	36	6	13.2	174.24
4	18	29	-11	-3.8	14.44
5	11	10	1	8.2	67.24
Sum	220	256	-36	0	460.8
Mean	44	51.2	-7.2	0	92.16

s = sqrt [ (\sum (d_i - d)² / (n - 1) ]

s = 10.73313

SE = s / sqrt(n)

S.E = 4.8

DF = n - 1 = 5 -1

D.F = 4

D)

t = [ (x₁ - x₂) - D ] / SE

t = - 1.50

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

E) Rejection region is p-value < 0.05.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 4 degrees of freedom is more extreme than 1.50; that is, less than - 1.50 or greater than 1.50.

F) Thus, the P-value = 0.208.

Interpret results. Since the P-value (0.208) is greater than the significance level (0.05), hence we failed to reject the null hypothesis.

G)

Do not reject H₀.  From the above test we have sufficient evidence to conclude that two types of UAVs perform the same.