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.

Answer 1

Answer:

There is no enough evidence that the probabilities of success for the two binomial experiments differ.

Step-by-step explanation:

The null and alternative hypothesis are:

$H_0: p_2-p_1=0\\\\H_a: p_2-p_1\neq0$

The significance level is 0.05.

The proportion of the first experiment is p_1=30/75=0.4.

The proportion of the second experiment is p_2=50/100=0.5.

The difference between proportions is

$\Delta p=p_2-p_1=0.5-0.4=0.1$

The standard deviation of the difference between the proportion is:

$\sigma=\sqrt{\frac{p_2(1-p_2)}{n_2}+\frac{p_1(1-p_1)}{n_1} } \\\\ \sigma=\sqrt{\frac{0.5*0.5}{100}+\frac{0.4*0.6}{75} } \\\\ \sigma=\sqrt{0.0057}=0.075$

Then, the z-statistic is:

$z=\frac{\Delta p}{\sigma}=\frac{0.1}{0.075} =1.33$

The p-value for this two-sided test is P(z>1.33)=0.09. This is bigger than the significance level, so the effect is not significant.

There is no enough evidence that the probabilities of success for the two binomial experiments differ.