3 Comparing two proportions23 Comparing two proportionsTwo drugs are compared to find out if the outcome of one of them is better thanthe other, or they are essentially equivalent. The data is as follows:drug 1drug 2patients tested40patients improved31Set up a test with the Null Hypothesis stating that there is not a significantdifference in outcome between the two.3525Note As discussed in the Lecture Note on Chapter 21, since we don't have ahypothesis about the two proportion separately, we will need to use theobserved proportions to set up the standard deviation. We can also usethe "pooled variance" instead. 1 Estimating ProportionsThe quality control people at your company have tested a sample of 450 widgetsand found that 23 were defective.What is your interval estimate (confidence interval) for the average propor-tion of defective widgets (choose your confidence level)?2 Setting Up A Test On How "Random" the Output of aShuffling Machine Is.A gambling establishment bought a new card shuffling machine. To do a simpletest for how "random” the card sequence will be after shuffling, they ran themachine a number of times, and recorded how many times an ace turned outon top of the card stack. Here are four (simulated) experiments, finding theproportion of shuffles that resulted in an ace as the first card. Since there are 4aces in 52 cards, the theoretical probability is 4/52 = ¹/13 ≈ 0.0769 Test for thisvalue of p, and compare the results when doing 100, 1000, and 10,000 shuffles(machines can do that). Note that all three simulations were done with thesame "true" p, so representing the same shuffler.Number of shuffles Times an ace came on top100100010000789832Note Treat these as three separate experiments, as we check how different sam-ple sizes may affect our test. Do not combine them in a single testBonus Using the critical/acceptance region approach (rather then only statingthe p-value), calculate the power, or, equivalently, the Error of Type II,as a function of one or more possible values for p (for example, p = 0.09,but feel free to choose any alternate hypothesis you like). Note that thiscan (and should) be done before performing the test.

Question

3 Comparing two proportions23 Comparing two proportionsTwo drugs are compared to find out if the outcome of one of them is better thanthe other, or they are essentially equivalent. The data is as follows:drug 1drug 2patients tested40patients improved31Set up a test with the Null Hypothesis stating that there is not a significantdifference in outcome between the two.3525Note As discussed in the Lecture Note on Chapter 21, since we don't have ahypothesis about the two proportion separately, we will need to use theobserved proportions to set up the standard deviation. We can also usethe &#34;pooled variance&#34; instead. 1 Estimating ProportionsThe quality control people at your company have tested a sample of 450 widgetsand found that 23 were defective.What is your interval estimate (confidence interval) for the average propor-tion of defective widgets (choose your confidence level)?2 Setting Up A Test On How &#34;Random&#34; the Output of aShuffling Machine Is.A gambling establishment bought a new card shuffling machine. To do a simpletest for how &#34;random” the card sequence will be after shuffling, they ran themachine a number of times, and recorded how many times an ace turned outon top of the card stack. Here are four (simulated) experiments, finding theproportion of shuffles that resulted in an ace as the first card. Since there are 4aces in 52 cards, the theoretical probability is 4/52 = ¹/13 ≈ 0.0769 Test for thisvalue of p, and compare the results when doing 100, 1000, and 10,000 shuffles(machines can do that). Note that all three simulations were done with thesame &#34;true&#34; p, so representing the same shuffler.Number of shuffles Times an ace came on top100100010000789832Note Treat these as three separate experiments, as we check how different sam-ple sizes may affect our test. Do not combine them in a single testBonus Using the critical/acceptance region approach (rather then only statingthe p-value), calculate the power, or, equivalently, the Error of Type II,as a function of one or more possible values for p (for example, p = 0.09,but feel free to choose any alternate hypothesis you like). Note that thiscan (and should) be done before performing the test.

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The quality control people at your company have tested a sample of 450 widgets and found that 23 were defective. What is your interval estimate (confidence interval) for the average propor- tion of defective widgets (choose your confidence level)?