Calculate a 95 percent confidence interval for µ d = µ 1 – µ 2 . Can we be 95 pe
ID: 3150667 • Letter: C
Question
Calculate a 95 percent confidence interval for µd = µ1 – µ2. Can we be 95 percent confident that the difference between µ1 and µ2 is greater than 0? (Round your answers to 2 decimal places.)
Test the null hypothesis H0: µd = 0 versus the alternative hypothesis Ha: µd 0 by setting equal to .10, .05, .01, and .001. How much evidence is there that µd differs from 0? What does this say about how µ1 and µ2 compare? (Round your answer to 3 decimal places.)
The p-value for testing H0: µd < 3 versus Ha: µd > 3 equals .1316. Use the p-value to test these hypotheses with equal to .10, .05, .01, and .001. How much evidence is there that µd exceeds 3? What does this say about the size of the difference between µ1 and µ2? (Round your answer to 3 decimal places.)
(a)Calculate a 95 percent confidence interval for µd = µ1 – µ2. Can we be 95 percent confident that the difference between µ1 and µ2 is greater than 0? (Round your answers to 2 decimal places.)
Explanation / Answer
a)
Note that
Margin of Error E = t(alpha/2) * s / sqrt(n)
Lower Bound = X - t(alpha/2) * s / sqrt(n)
Upper Bound = X + t(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.025
X = sample mean = 4.1
t(alpha/2) = critical t for the confidence interval = 2.010634758
s = sample standard deviation = 6.8
n = sample size = 49
df = n - 1 = 48
Thus,
Margin of Error E = 1.95318805
Lower bound = 2.14681195
Upper bound = 6.05318805
Thus, the confidence interval is
( 2.14681195 , 6.05318805 )[ANSWER]
YES, as this whole interval is greater than 0. [ANSWER]
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b)
Formulating the null and alternative hypotheses,
Ho: u = 0
Ha: u =/ 0
As we can see, this is a two tailed test.
df = n - 1 = 48
Getting the test statistic, as
X = sample mean = 4.1
uo = hypothesized mean = 0
n = sample size = 49
s = standard deviation = 6.8
Thus, t = (X - uo) * sqrt(n) / s = 4.220588235
Also, the p value is, as this is two tailed,
p = 0.000107681
As this P is less than all 0.10, 0.05, 0.01, and 0.001, WE REJECT HO AT ALPHA EQUALS ALL TEST VALUES. [ANSWER]
EXTREMELY STRONG evidence that u1 differs from u2. [ANSWER]
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c)
Formulating the null and alternative hypotheses,
Ho: u <= 3
Ha: u > 3
As we can see, this is a right tailed test.
df = n - 1 = 48
Getting the test statistic, as
X = sample mean = 4.1
uo = hypothesized mean = 3
n = sample size = 49
s = standard deviation = 6.8
Thus, t = (X - uo) * sqrt(n) / s = 1.132352941 [ANSWER, T VALUE]
As P = 0.1316 is greater than all the alpha values given,
Reject H0 at equal to NO TEST VALUES. [ANSWER]
NO evidence that µ1 and µ2 differ by more than 3. [ANSWER]
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