
To construct:
Each specified confidence interval and interpret the interval.

Answer to Problem 20E
Solution:
The confidence interval is (0.4126, 2.4637).
Explanation of Solution
Confidence interval is used to compare two population variances, estimate the value of the ratio or fraction, σ21σ22.
1. If the confidence interval contains 1 that means the two population variances are unequal at a particular level of confidence.
2. If both endpoints of the interval are greater than 1 it concludes that σ21 is larger than σ22.
3. If both endpoints of the interval are less than 1 it concludes that σ21 is smaller than σ22.
Given Information:
From the given information, Amy’s practice run is n1=16 and Veronica’s practice run is n2=15. Their population variances are s21=2.560 and s22=2.519 at 90% level of confidence.
Formula used:
The formula to calculate the confidence interval is,
(s21s22.1Fα2)<σ21σ22<(s21s22.1F(1−α2)).
The formula to calculate the point estimate is,
s21s22
Where, s1 and s2 are the sample variances for the respective samples.
For the given level of confidence c, the level of significance is α=1−c.
The right and left endpoints for F-distribution are α2 and 1−α2 respectively. The critical values are Fα2 and F1−α2 that can be determined in the F-distribution table using the column for df1=n1−1 and row for df2=n2−1.
Calculation:
Substitute 2.560 for s21 and 2.519 for s22 in the point estimate formula as,
s21s22=2.5602.519=1.0163
At 90% level of confidence or c=0.90, the level of significance is,
α=1−0.90=0.10
The right endpoint for F-distribution is α2. Substitute 0.10 for α in α2.
α2=0.102=0.05
Similarly the left endpoint for F-distribution is 1−α2. Substitute 0.10 for α in α2.
1−α2=1−0.05=0.95
The degrees of freedom are,
df1=n1−1……(1)
And,
df2=n2−1……(2)
The sample sizes are given as n1=16 and n2=15.
Substitute 16 for n1 in equation (1).
df1=n1−1=16−1=15
Substitute 15 for n2 in equation (2).
df2=n2−1=15−1=14
For the column of df1=15 and the row of df2=14 in the F-distribution table under the area of α2, the value of Fα2 is 2.4630.
For the column of df1=15 and the row of df2=14 in the F-distribution table under the area of 1−α2, the value of F1−α2 is 0.4125.
The formula to calculate the confidence interval is,
(s21s22.1Fα2)<σ21σ22<(s21s22.1F(1−α2)).
Substitute 1.0163 for s21s22, 2.4630 for Fα/2 and 0.4125 for F(1−α/2) in the above formula of confidence interval,
Confidence interval=(1.0163×12.4630)<σ21σ22<(1.0163×10.4125)=(0.4126)<σ21σ22<(2.4637)
Thus, the confidence interval is (0.4126, 2.4637).
Interpretation:
Since the value 1 is in the interval (0.4126, 2.4637), the data do not provide evidence at 90% level of confidence that, the population variances of Amy’s run time and Veronika’s run time are unequal.
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Chapter 9 Solutions
Beginning Statistics, 2nd Edition
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