Suppose you are studying the monthly amounts that seventh- and eighth-grade boys
ID: 3218221 • Letter: S
Question
Suppose you are studying the monthly amounts that seventh- and eighth-grade boys and girls spend on entertainment such as movies, music CDs, and candy. Representative samples of children within a certain school district were selected, and children were asked about their spending habits. The following results were obtained: 7th (a) Test whether the four groups differ with respect to their mean spending amounts. (b) Follow up on your analysis in (a) if you find differences. In particular, assess whether there are differences in the mean spending amounts of seventh- and eighth-grade boys and in the mean spending amounts of seventh- and eighth-grade girls.Explanation / Answer
a. State the hypotheses:
H0:mu1=mu2=mu3=mu4 (mean spending amounts are equal for all four grades)
H1:atleast two of the means differ.
Assumptions:
Model: Independent random.
Level of measurement: Interval-ratio.
Population variances are equal.
Sampling distribution: Normal.
Test statistic.
The grand mean, xbar=(x1bar+x2bar+x3bar+x4bar)/(n1+n2+n3+n4)=(20.1+23.2+19.6+25)/(30+25+30+25)=0.7991
Sum of sqaure between, SSB=n1(x1bar-xbar)^2+n2(x2bar-xbar)^2+n3(x3bar-xbar)^2+n4(x4bar-xbar)^2
=30(20.1-0.7991)^2+25(23.2-0.7991)^2+30(19.6-0.7991)^2+25(25-0.7991)^2
=48967.0545
Mean sqaure between, MSB=SSB/k-1, where, k is number of groups.
=48967.0545/(4-1)
=16322.35
Sum of sqaures within, SSW=(n1-1)s1^2+(n2-1)s2^2+(n3-1)s3^2+(n4-1)s4^2
=(30-1)*6^2+(25-1)*5.6^2+(30-1)*5.3^2+(25-1)*7^2
=3787.25
Mean sum of sqaure within, MSE=SSE/n-k=3787.25/(110-4)=35.73
F=MSB/MSW=16322.35/3787.25=4.31
the p value at (3, 106) degrees of freedom is: 0.00655.
Conclusion: Per rule, reject H0, if p value is less than alpha=0.05. Here, p value is less than 0.05, therefore, reject H0 to conclude that atleast one graders' mean spending amount is significantly different from another graders'.
b. Compute the end points for Tukey's 95% c.i for multiple comaprison method.
The 95% c.i for mu1-mu2 is as follows:
(x1bar-x2bar)+-qalpha/sqrt 2*s sqrt[1/n1+1/n2], where, s=sqrt MSE, qalpha at alpha=0.05, k=4, and v=110-4=106 is 3.68
=(20.1-23.2)+-3.68/sqrt 2* sqrt 35.73*sqrt(1/30+1/25)
=(-7.31, 1.11)
The confidence interval contain 0, therefore, there is no significant difference in mean spending amount sbetween 7th grade boys and 8th grade boys.
Similarly compute endpoints for 95% c.i for mu3-mu4.
(x3bar-x4bar)+-qalpha/sqrt 2*s*sqrt(1/n3+1/n4)
=(19.6-25)+-3.68/sqrt 2*sqrt 35.73*sqrt(1/30+1/25)
=(-9.61, -1.19)
The confidence interval does not contain 0, therefore, there is significant difference in mean spending amount between 7th grade girls and 8th grade girls.
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