Question

\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 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.The standard normal. The number of trials is sufficiently large.

(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 statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(g) Interpret the results.

Fail to 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 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.

Answer 1

(a)

Pooled probability of success for the two experiments = (r1 + r2) / (n1 + n2)

= (30 + 50) / (75 + 100)

= 0.457

(b)

Each sample includes at least 10 successes and 10 failures. Thus, the distribution does the sample test statistic follow

The standard normal. The number of trials is sufficiently large.

(c)

H₀: p₁ = p₂; H₁: p₁ ≠ p₂

(d)

p̂₁ = r1 / n1 = 30 / 75 = 0.4

p̂₂ = r2 / n2 = 50 / 100 = 0.5

p̂₁ - p̂₂ = 0.4 - 0.5 = -0.1

Standard error (SE) of the sampling distribution difference between two proportions.

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] } = sqrt{ 0.457 * ( 1 - 0.457 ) * [ (1/75) + (1/100) ] } = 0.076

Test statistic, z = |p̂₁ - p̂₂ | / SE = 0.1 / 0.076 = 1.316

(e)

For two-tail test, P-value = 2 * P[z > 1.316] = 0.1882

(f)

Since p-value is greater than α = 0.05 , 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.