Question I A pooled variance ttest problem A sporting goods

Question I – A pooled variance t-test problem

A sporting goods manufacturer wants to compare the distance traveled by two different designs of golf ball. The manufacturer wants to market the two different designs of golf balls at two different price levels, better distance golf balls sell at a higher markup. A random sample of golf balls of each design was selected and brought to the local golf course for the club professional to test. The order in which the balls were hit with the same club from the first tee was randomized so that the golf pro did not know which type of ball was being hit. All balls were hit in a short space of time, during which the environmental conditions remained constant. You may assume that the variances for these two designs are equal. The results (distance in yards) for the two designs were as follows:

Golfballs

Design 1

Design 2

239.4

226.4

232.2

226.9

234.6

211.7

224.7

231.9

227.1

230.0

229.9

224.7

229.9

233.3

237.7

215.0

223.1

227.1

239.6

235.5

a)      State the null and alternative hypotheses for this problem, including the appropriate decision rule, if you want to analyze the differences in driving distances between the two golf ball designs using = 0.05.

b)      What is the appropriate statistical test to use to test your hypotheses? Justify your choice.

c)       State the formula (or computer steps) to compute the test statistic. Provide the test statistic.

d)      Based on your findings, what is your statistical conclusion?

e)      What recommendation would you make to the manufacturer regarding the two golf ball designs?

Solution

Set Up Hypothesis
Null, There Is NoSignificance between them Ho: u1 = u2
Alternative,differences in driving distances between the two golf ball designs H1: u1 != u2
Test Statistic
X (Mean)=232.15; Standard Deviation (s.d1)=5.736
Number(n1)=11
Y(Mean)=225.22; Standard Deviation(s.d2)=7.314
Number(n2)=9
Value Pooled variance S^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
S^2 = (10*32.9017 + 8*53.4946) / (20- 2 )
S^2 = 42.0541
we use Test Statistic (t) = (X-Y)/Sqrt(S^2(1/n1+1/n2))
to=232.15-225.22/Sqrt((42.0541( 1 /11+ 1/9 ))
to=6.93/2.9148
to=2.3776
| to | =2.3776
Critical Value
The Value of |t | with (n1+n2-2) i.e 18 d.f is 2.101
We got |to| = 2.3776 & | t | = 2.101
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != 2.3776 ) = 0.0281
Hence Value of P0.05 > 0.0281,Here we Reject Ho

We have evidence that differences in driving distances between the two golf ball designs