**Hypothesis Testing for Binomial Experiments**For one binomial experiment, \( n_1 = 75 \) binomial trials produced \( x_1 = 30 \) successes. For a second independent binomial experiment, \( n_2 = 100 \) binomial trials produced \( x_2 = 50 \) 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.)**\[ \hat{p} = \]**(b) Check Requirements: What distribution does the sample test statistic follow? Explain.**- The standard normal. We assume the population distributions are approximately normal.- The standard normal. The number of trials is sufficiently large.- The Student's t. We assume the population distributions are approximately normal.- The Student's t. The number of trials is sufficiently large.**(c) State the hypotheses.**- \( H_0: p_1 = p_2; \, H_1: p_1 < p_2 \)- \( H_0: p_1 = p_2; \, H_1: p_1 > p_2 \)- \( H_0: p_1 = p_2; \, H_1: p_1 \neq p_2 \)- \( H_0: p_1 < p_2; \, H_1: p_1 = p_2 \)**(d) Compute \(\hat{p}_1 - \hat{p}_2\):**\[ \hat{p}_1 - \hat{p}_2 = \]Compute the corresponding sample distribution value. (Test the difference \( p_1 - p_2 \). 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.)**\[ \text{P-value} = \]**(f) Conclude the test.**- At the \(\alpha = 0.05\) level, we reject the null hypothesis and conclude the data are statistically significant.- At the \(\alpha = 0.05\) level, we reject the null hypothesis and

Given, sample 1 : Binomial trials = n1 = 75 Number of successes = r1 = 30 Sample proportion = p^1 = r1n1 =3075=0.4 sample 2 : Binomial trials = n2 = 100 Number of successes = r2 = 50 Sample proportion = p^2 = r2n2 =50100=0.5 a) We have to find pooled proportion. p = r1+r2n1+n2 p = 30+5075+100 p= 0.457143 p= 0.46 b) The standard normal. The number of trails is sufficiently large. c) Claim : The probabilities of success for the two binomial experiments differ , i.e. p1 ≠p2 So the hypothesis is , H0 : p1=p2 H1 : p1≠p2 Two tailed test. d) p^1-p^2 =0.4-0.5 =-0.1 Sampling distribution value ( test statistic ) , z=p^1-p^2p(1-p)1n1+1n2 z=0.4-0.50.457143(1-0.457143)175+1100 z = -1.3141437 z = -1.31 d) P-value : P-value for this two tailed test is , p-value = 2*P(Z<z) = 2*P(Z< -1.3141437) Using Excel function , =NORMSDIST(z)" P(Z> -1.3141437) = NORMSDIST(-1.3141437) = 0.094399 So, p-value = 2*0.094399 = 0.18879785 P-value = 0.1888 e) Decision about null hypothesis : Decision rule : 1) Reject null hypothesis (H0) if p-value less than significance level (α ) 2) Otherwise fail to reject null hypothesis (H0) Let , Significance level (α) =5% = 0.05 It is observed that p-value (0.1888) is greater than α = 0.05 So, Fail to reject null hypothesis f) At the =0.05 level we 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 proportion of the probabilities of success for the two binomial experiments differ.

Math

Statistics

For one binomial experiment, n, - 75 binomial trials produced r, - 30 successes. For a second independent binomial experiment, n - 100 binomial trials produced r, - 50 successes. At the 5% level of significance, test the claim that the probabilities of success for the two binomial experiments differ. A USE SALT (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. O The standard normal. We assume the population distributions are approximately normal. O The standard normal. The number of trials is sufficiently large. O The Student's t. We assume the population distributions are approximately normal. O The Student's t. The number of trials is sufficiently large. (c) State the hypotheses. O Hg: P - Pai HiP> P2 O Hg: P - Pai HạiPz * P2 O Hạ: P < Pzi HP"P2 (d) Compute p, - P2 Compute the corresponding sample distribution value. (Test the difference p, -P,. 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.) () Conclude the test. O At the a- 0.0S level, we reject the null hypothesis and conclude the data are statistically significant. O At the a- 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. O At the a- 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. O At the a = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (9) Interpret the results. O Reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ. O Reject the null hypothesis, there is insufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ. O Fail to reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ. O Fail to reject the null hypothesis, there is insufficient evidence that the probabilities of success for the two binomial experiments differ.