For one binomial experiment,
n1 = 75
binomial trials produced
r1 = 30
successes. For a second independent binomial experiment,
n2 = 100
binomial trials produced
r2 = 50
successes. At the 5% level of significance, test the claim that the probabilities of success for the two binomial experiments differ.
(a) Compute the pooled probability of success for the
two experiments. (Round your answer to three decimal
places.)
(b) Check Requirements: What distribution does the sample test statistic follow? Explain.
The standard normal. The number of trials is
sufficiently large.The Student's t. The number of trials is
sufficiently large. The Student's t.
We assume the population distributions are approximately normal.The
standard normal. We assume the population distributions are
approximately normal.
(c) State the hypotheses.
H0: p1 =
p2; H1: p1 ≠
p2H0: p1 =
p2; H1: p1
>
p2 H0:
p1 < p2;
H1: p1 =
p2H0: p1 =
p2; H1: p1
< p2
(d) Compute p̂1 - p̂2.
p̂1 - p̂2 =
Compute the corresponding sample distribution value.
(Test the difference p1 − p2.
Do not use rounded values. Round your final answer to two decimal
places.)
(e) Find the P-value of the sample test
statistic. (Round your answer to four decimal places.)
(f) Conclude the test.
At the α = 0.05 level, we reject the null hypothesis
and conclude the data are statistically significant.At the α = 0.05
level, we reject the null hypothesis and conclude the data are not
statistically significant. At the α = 0.05
level, we fail to reject the null hypothesis and conclude the data
are not statistically significant.At the α = 0.05 level, we fail to
reject the null hypothesis and conclude the data are statistically
significant.
(g) Interpret the results.
Fail to reject the null hypothesis, there is insufficient evidence that the probabilities of success for the two binomial experiments differ.Reject the null hypothesis, there is insufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ. Reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.Fail to reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.
(b) The standard normal. The number of trials is sufficiently
large.
(c) H0: p1 =
p2; H1:
p1 ≠ p2.
(f) At the α = 0.05 level, we fail to reject the null hypothesis
and conclude the data are not statistically significant.
(g) Fail to reject the null hypothesis, there is insufficient
evidence that the probabilities of success for the two binomial
experiments differ.
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