In a binomial experiment with n trials and probability of success p, if np(1-p) V the binomial random variable X is approximately normal with Px and
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- Q. Consider the random variable N which counts the number of delays of a given flight in a year. Empirical data suggests that the expected value of N is 10 and its variance is 4. In a specific year 27 delays were observed. (a) Would you classify this event as atypical? (b) Give estimate to the probability of observing 30 delays or more.Suppose that you want to perform a hypothesis test based on independent random samples to compare the means of two populations. You know that the two distributions of the variable under consideration have the same shape and may be normal. You take the two samples and find that the data for one of the samples contain outliers. Which procedure would you use? Explain your answer.The new study takes a random sample of 14 vaccinated high school children. Let x be the number of children in the sample who contract the flu. The p-value for the test can be calculated from a Binomial distribution using P(X≤x). The maximum number of children who can contract the flu to give evidence against the null hypothesis in Question 3 at the 5% level is 1 2 3 5 Question 5 Suppose the actual population proportion of vaccinated high school children who contract the flu is 30%. For a random sample of 14 vaccinated high school children and based on your answer to Question 4, the probability of making a Type II error is 0.3637 0.6448 0.8392 0.8631 Question 6 The new study also carried out a test to determine whether the population proportion of unvaccinated school children contracting winter flu was higher than the population proportion of vaccinated school children. The Z test statistic to test this belief is found to be 1.874. The corresponding p-value is 0.0305 0.1212 0.3036…
- Let a be a random variable representing the percentage of protein content for early bloom alfalfa hay. The average percentage protein content of such early bloom alfalfa should be u = 17.2%. A farmer's co-op is thinking of buying a large amount of baled hay but suspects that the hay is from a later summer cutting with lower protein content. A small amount of hay was removed from each bale of a random sample of 50 bales. The average protein content from the samples was determined by a local agricultural college to be -15.8% with a sample standard deviation of S = 5.3%- At a = .05, does this hay have lower average protein content than the early bloom alfalfa?Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. Refer to screenshot. (a) Suppose n = 27 and p = 0.23. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.) for the second part the options are: (Yes, no) p̂ (can, cannot) be approximated by a normal random variable because (np exceeds, np and nq do not exceed, nq exceeds, nq does not exceed, both np and nq exceed, np does not exceed) ___________ (fill in blank). What are the values of ?p̂ and ?p̂? (Use 3 decimal places.)1. Suppose you flip a fair coin infinitely many times. Let K₁, be the number of times the pattern HHHH occurs in the first n flips. What are the expected value and variance of Kn? Is the variance larger or smaller than the variance of a Binomial random variable with the same mean? Can you explain the difference intuitively? For n = 100, simulate Kn 3000 times and make a histogram of the proportion of times each possible outcome actually occurred in your simulation. Plot this histogram versus the probability mass function of the Binomial random variable with the same mean.
- Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. (a) Suppose n = 30 and p = 0.25. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.) np = 7.5 ng = Yes be approximated by a normal random variable because both np and nq exceed v can What are the values of , and o? (Use 3 decimal places.) (b) Suppose n = 25 and p = 0.15. Can we safely approximate p by a normal distribution? Why or why not? No ,p cannot be approximated by a normal random variable because np does not exceed (c) Suppose n = 51 and p = 0.24. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.) np = nq = Yes be approximated by a normal random variable because both np and nq exceed can What are the values of u, and o,? (Use 3 decimal places.) LOA computer repair shop has two work centers. The first center examines the computer to see what is wrong and the second center repairs the computer. Let and be random variables representing the lengths of time in minutes to examine a computer () and to repair a computer (). Assume and are independent random variables. Long-term history has shown the following mean and standard deviation for the two work centers: Examine computer, : = 27.3 minutes; = 7.5 minutes Repair computer, : = 90.1 minutes; = 15.3 minutes Let be a random variable representing the total time to examine and repair the computer. Suppose it costs $1.80 per minute to examine the computer and $2.83 per minute to repair the computer. Then is a random variable representing the service charges (without parts). Compute the mean and standard deviation of V. Round your answer to the nearest tenth.Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. (b) Suppose n = 25 and p = 0.15. Can we safely approximate p̂ by a normal distribution? Why or why not? (Yes/No) , p̂ (can/cannot) be approximated by a normal random variable because ____(np and nq do not exceed, nq exceeds, nq does not exceed, np does not exceed, nq does not exceed, np exceeds, both np and nq exceed) ________ (fill in blank) (c) Suppose n = 43 and p = 0.20. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.) np = ??? nq = ??? (Yes/No) , p̂ (can/cannot) be approximated by a normal random variable because ____(np and nq do not exceed, nq exceeds, nq does not exceed, np does not exceed, nq does not exceed, np exceeds, both np and nq exceed) ________ (fill in blank) What are the values of ?p̂ and ?p̂? (Use 3 decimal places.)
- Each scenario shows the data used for a randomisation test on a difference in two proportions. Both have a sample size of n = 30, with half being assigned to Treatment A and half to Treatment B. Scenario 1 has a difference in proportions of 0.4. Scenario 2 has a difference in proportions of 0.2. If we ran a randomisation test for each scenario, the one that would result in a smaller tail proportion in the resulting re-randomisation distribution is: Scenario 2 Scenario 1 Data Yes Yes Data Yes Yes 01 4 LO 5 02 7 No 03 No 0.2 11 →No 05 11 0.4 06 0.7 10 67 8 No DO 4 09 B B 10Suppose we are planning a clinical trial and expect a 20% success rate in the active group and a 10% success rate in the placebo group. We expect to enroll 100 participants in each group and are interested in the power of the study. (a) Perform a simulation study, and generate 100 participants from a binomial distribution with p = 0.2 and 100 participants from a binomial distribution with p = 0.1. Test to determine whether the observed sample proportion of successes are significantly different, using a two-sided test with a = 0.05. (b) Repeat the Problem (a) simulation 1000 times, and compute the proportion of simulations in which a significant difference is found (i.e., an estimate of the power of the study).Example 10.5.4 Out of 160 offsprings obtained due to cross-breeding bet- ween guinea pigs, 102 are red, 24 are black and 34 are white. According to a genetic model, the probabilities of red, black and white are 9/16, 3/16, 1/4, respectively. Test at 5% significance level if the data is consistent with the model. (For 2 degrees of freedom, P(x² > 5.99) = 0.05.)
