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10-4 PAIRED t-TEST 349

Brand 1: 275, 286, 287, 271, 283, 271, 279, 275, 263, 267 clubmakers are tested and the coefficient of restitution meas-
Brand 2: 258, 244, 260, 265, 273, 281, 271, 270, 263, 268 ured. The data follow:

(a) Is there evidence that overall distance is approximately Club 1: 0.8406, 0.8104, 0.8234, 0.8198, 0.8235, 0.8562,
normally distributed? Is an assumption of equal variances 0.8123, 0.7976, 0.8184, 0.8265, 0.7773, 0.7871
justified? Club 2: 0.8305, 0.7905, 0.8352, 0.8380, 0.8145, 0.8465,
(b) Test the hypothesis that both brands of ball have equal 0.8244, 0.8014, 0.8309, 0.8405, 0.8256, 0.8476
mean overall distance. Use 0.05.
(c) What is the P-value of the test statistic in part (b)? (a) Is there evidence that coefficient of restitution is approxi-
(d) What is the power of the statistical test in part (b) to detect mately normally distributed? Is an assumption of equal
a true difference in mean overall distance of 5 yards? variances justified?
(e) What sample size would be required to detect a true dif- (b) Test the hypothesis that both brands of ball have equal
ference in mean overall distance of 3 yards with power of mean coefficient of restitution. Use 0.05.
approximately 0.75? (c) What is the P-value of the test statistic in part (b)?
(f) Construct a 95% two-sided CI on the mean difference in (d) What is the power of the statistical test in part (b) to detect
overall distance between the two brands of golf balls. a true difference in mean coefficient of restitution of 0.2?
10-32. In Example 9-6 we described how the “spring-like (e) What sample size would be required to detect a true dif-
effect” in a golf club could be determined by measuring the ference in mean coefficient of restitution of 0.1 with
coefficient of restitution (the ratio of the outbound velocity to power of approximately 0.8?
the inbound velocity of a golf ball fired at the clubhead). (f) Construct a 95% two-sided CI on the mean difference in co-
Twelve randomly selected drivers produced by two efficient of restitution between the two brands of golf clubs.

10-4 PAIRED t-TEST

A special case of the two-sample t-tests of Section 10-3 occurs when the observations on
the two populations of interest are collected in pairs. Each pair of observations, say (X 1j ,
X 2j ), is taken under homogeneous conditions, but these conditions may change from one
pair to another. For example, suppose that we are interested in comparing two different
types of tips for a hardness-testing machine. This machine presses the tip into a metal spec-
imen with a known force. By measuring the depth of the depression caused by the tip, the
hardness of the specimen can be determined. If several specimens were selected at random,
half tested with tip 1, half tested with tip 2, and the pooled or independent t-test in Section
10-3 was applied, the results of the test could be erroneous. The metal specimens could
have been cut from bar stock that was produced in different heats, or they might not
be homogeneous in some other way that might affect hardness. Then the observed differ-
ence between mean hardness readings for the two tip types also includes hardness differ-
ences between specimens.
A more powerful experimental procedure is to collect the data in pairs—that is, to make
two hardness readings on each specimen, one with each tip. The test procedure would then
consist of analyzing the differences between hardness readings on each specimen. If there is
no difference between tips, the mean of the differences should be zero. This test procedure is
called the paired t-test.
Let (X 11 , X 21 ), (X 12 , X 22 ), p , (X 1n , X 2n ) be a set of n paired observations where we assume
2
that the mean and variance of the population represented by X 1 are 1 and , and the mean
1
2
and variance of the population represented by X 2 are 2 and . Define the differences be-
2
tween each pair of observations as D j X 1j X 2j , j 1, 2, p , n. The D j ’s are assumed to be
normally distributed with mean
E1X X 2 E1X 2 E1X 2 2
1
D
2
1
2
1