(a) Calculate a 95 percent confidence interval for µ d = µ 1 – µ 2 . (Round your
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Question
(a) Calculate a 95 percent confidence interval for µd = µ1 – µ2. (Round your answers to 2 decimal places.)
Confidence interval = [ , ]
(b) 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?
(c) The p-value for testing H0: µd < 3 versus Ha: µd > 3 equals 0.0256. 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 p-value answer to 4 decimal places.)
t = Reject H0 at ? equal to (Click to select)all test valuesno test values0.10.1 and 0.0010.05 (Click to select)extremely strongstrongnovery strongsome evidence that µ1 differs from µ2. a 35Explanation / Answer
Solution:
a) CI formula is given by:
(x-bar - t* (sample sd), x-bar + t* (sample sd) ) =
(5 - t* (7/sqrt(49)), 5+ t* (7/sqrt(49)))
where t* is the critical value for 95% area under the t-distribution. Since the sample is large enough, this is roughly equal to z*, the critical value of normal distribution = 1.96,
so Your CI is: (3.04, 6.96)
since the CI does not contains 0, neither does it contain any negative values, so we can be 95% confident that the difference between u1 and u2 is greater than zero.
b) Your test-statistics t = (5-0)/(7/sqrt(49)) = 5
again, the sample is large enough to be approximated by normal,
the critical value z for alpha=0.10 is 1.645, and z for alpha = 0.001 is 3.32
since your test statistics is much larger than both critical value, we reject the null hypothesis in favor of Ha.
c) You are given the p-value = 0.0256 which is between 0.10 and 0.001. So we would reject if alpha = 0.10, but retain if alpha = 0.001. What this means is that we are 97.44% confident that ud is greater than 3. So it is pretty likely that the difference between u1 and u2 is greater than 3.
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