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- If X1, X2, ... , Xn constitute a random sample of size n from an exponential population, show that X is a consis-tent estimator of the parameter θ.If the alleles A and B of the cystic fibrosis gene occur in a population with frequencies p and 1 - p (where pis between 0 and 1), then the frequency of heterozygous carriers (carriers with both alleles) is 2p(l - p). Which value of p gives the largest frequency of heterozygous carriers?A certain company produces fidget spinners with ball bearings made of either plastic or metal. Under standard testing conditions, fidget spinners from this company with plastic bearings spin for an average of 2.7 minutes, while those from this company with metal bearings spin for an average of 4.2 minutes. A random sample of three fidget spinners with plastic bearings is selected from company stock, and each is spun one time under the same standard conditions; let x¯1x¯1 represent the average spinning time for these three spinners. A random sample of seven fidget spinners with metal bearings is selected from company stock, and each is likewise spun one time under standard conditions; let x¯2x¯2 represent the average spinning time for these seven spinners. What is the mean μ(x¯1−x¯2)μ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2x¯1−x¯2 ? 3(2.7)−7(4.2)=−21.33(2.7)−7(4.2)=−21.3 A 3−7=−43−7=−4 B 2.7−4.2=−1.52.7−4.2=−1.5 C…
- Show that (X+1)/(n+2) is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?Resistors labeled as 100 Ω are purchased from two different vendors. The specification for this type of resistor is that its actual resistance be within 5% of its labeled resistance. In a sample of 180 resistors from vendor A, 150 of them met the specification. In a sample of 270 resistors purchased from vendor B, 233 of them met the specification. Vendor A is the current supplier, but if the data demonstrate convincingly that a greater proportion of the resistors from vendor B meet the specification, a change will be made. a) State the appropriate null and alternate hypotheses. b) Find the P-value. c) Should a change be made?Resistors labeled as 100 Ω are purchased from two different vendors. The specification for this type of resistor is that its actual resistance be within 5% of its labeled resistance. In a sample of 180 resistors from vendor A, 149 of them met the specification. In a sample of 270 resistors purchased from vendor B, 233 of them met the specification. Vendor A is the current supplier, but if the data demonstrate convincingly that a greater proportion of the resistors from vendor B meet the specification, a change will be made. P-value?
- 1.- A study conducted in the automotive field states that more than 40% of vehicle engine failures are due to problems in the cooling system. To test this statement, a study is carried out on 70 vehicles and the critical region is defined as x < 26, where x is the number of vehicle engines that have problems in the cooling system. (use the normal approximation)a) Evaluate the probability of making a type I error, assuming that p = 0.4.b) Evaluate the probability of committing a type II error, for the alternative p = 0.3.In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signicantly dierent from$21, which is the average bill for current AT&T customers (d) Using 5% signicance level, calculate the critical values of the test. Is the null hypothesis rejected?(e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 andH1 : µ 6= 19. What…In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signicantly dierent from$21, which is the average bill for current AT&T customers.(a) The statistician wants to formulate hypotheses in favor of AT&T. State the hypotheses.(b) Calculate the t-statistic. Under what signicance level, among 1%, 5% and 10%, is the null hypothesis rejected?(c) Calculate the…
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?In a clinical study, a random sample of 540 participants agree to have their blood drawn, which is to be examined for the presence of antibodies against a certain contagious disease. It is found in 22% of the blood samples, which experimenters hope to extrapolate to the general population. From this random sample, 10 participants' blood samples are selected at random. If X is the number of samples out of the 10 who have these antibodies, what can we say about X? A. The sample size is not large enough for us to approximate X using a normal distribution B.The expected value of X is 22 C. X can be approximated using a normal distribution in lieu of a binomial distribution D. X has a sampling distribution that is normalThe critical value for t for n = 10 for a one tail test at α = 5% LOS is