Setup: JP took two random samples from different populations. Sample 1: A random sample of size n1 = 15 is taken from an approximate normal population. From our sample, we find a variance of s1^2= 11.1672, and a mean of x¯1 = 115.1192. Sample 2: A second random sample of size n2 = 23 is taken from a different approximate normal population. From our second sample, we find a variance of s2^2 = 6.310136, and a mean of ¯x2 = 125.0925. Goal: We wish to test the hypothesis that µ1 − µ2 = −8, against the alternative µ1 − µ2 < −8 at a significance level of 2.9%. R Output: > t.test(data1,data2, mu = -8, var.equal = FALSE, alternative = "less", conf.level = 0.971) Welch Two Sample t-test data: data1 and data2 t = -1.955, df = 24.134, p-value = 0.03113 alternative hypothesis: true difference in means is less than -8 97.1 percent confidence interval: -Inf -7.964374 sample estimates: mean of x mean of y 115.1192 125.0925 Questions: (a) Why was the test run with var.equal = FALSE? (b) State the appropriate p-value, based on the output provided. (c) What is your decision for the hypothesis test (Reject or Not Reject H0)? (d) State your conclusion. (e) If you were doing this hypothesis test using critical regions instead (everything else being the same), would you end up with the same decision and conclusion? Why or why not?
Setup: JP took two random samples from different populations.
Sample 1: A random sample of size n1 = 15 is taken from an approximate normal
population. From our sample, we find a variance of s1^2= 11.1672, and a mean of x¯1 = 115.1192.
Sample 2: A second random sample of size n2 = 23 is taken from a different
approximate normal population. From our second sample, we find a variance of
s2^2 = 6.310136, and a mean of ¯x2 = 125.0925.
Goal: We wish to test the hypothesis that µ1 − µ2 = −8,
against the alternative µ1 − µ2 < −8 at a significance level of 2.9%.
R Output:
> t.test(data1,data2, mu = -8, var.equal = FALSE,
alternative = "less", conf.level = 0.971)
Welch Two Sample t-test
data: data1 and data2
t = -1.955, df = 24.134, p-value = 0.03113
alternative hypothesis: true difference in means is less than -8
97.1 percent confidence interval:
-Inf -7.964374
sample estimates:
mean of x mean of y
115.1192 125.0925
Questions:
(a) Why was the test run with var.equal = FALSE?
(b) State the appropriate p-value, based on the output provided.
(c) What is your decision for the hypothesis test (Reject or Not Reject H0)?
(d) State your conclusion.
(e) If you were doing this hypothesis test using critical regions instead (everything else
being the same), would you end up with the same decision and conclusion? Why or
why not?
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