Concept explainers
Refer to Exercise 3.147. Use the uniqueness of moment-generating
3.147 If Y has a geometric distribution with
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Chapter 3 Solutions
Mathematical Statistics with Applications
- A researcher conducts a hypothesis test using a sample of n = 20 with M = 34 and s 2 = 36 from an unknown population. What is the df value for the t statistic?arrow_forwardA researcher conducts a hypothesis test using a sample of n=20 with M =34 and s^2=36 from an unknown population.What is the df value for the t statistic?arrow_forwardA train’s arrival time at the station is T minutes past noon, where the PDF of T is given below. Given that the train hasn’t arrived by 12:05, what is the probability that it arrives by 12:10?arrow_forward
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)arrow_forwardUse the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.arrow_forwardUse the standard error to construct a approximate prediction interval for Y using an alpha of 5%. (Round your answer to 2 decimal places x x Fuel Economy of Selected Vehicles (n = 73, k = 4) Obs Vehicle CityMPG Length Width Weight ManTran 1 Acura TL 20 109.3 74.0 3968 0 2 Audi A5 22 108.3 73.0 3583 1 3 BMW 4 Series 428i 22 182.6 71.9 3470 0 4 BMW X1 sDrive28i 23 176.5 70.8 3527 0 5 Buice LaCrosse 18 196.9 73.1 3990 0 6 Buick Enclave 17 201.9 79.0 4724 0 7 Buick Regal 21 190.2 73.1 3692 0 8 Cadillac ATS 22 182.8 71.1 3315 0 9 Cadillac CTS 20 195.5 72.2 3616 0 10 Cadillac Escalade 14 202.5 79.0 5527 0 11 Chevrolet Camaro 1SS 16 190.6 75.5 3719 1 12 Chevrolet Cruze LS 26 181.0 70.7 3097 0 13 Chevrolet Impala LTZ 19 201.3 73.0 3800 0 14 Chevrolet Malibu 2LT 25 191.5 73.0 3532 0 15 Chevrolet Spark LS 31 144.7 61.0 2269 1 16 Chevrolet Suburban LTZ 15 224.0 80.5 5674 0 17 Chrysler 200 Touring LX 20 191.7 72.5…arrow_forward
- A data analytics firm hired by a Republican candidate running for political office wants to test whether its practices were successful in increasing voter turnout among registered Republicans. Suppose that data were collected for a random set of 10 cities, where each difference is calculated by subtracting the percent voter turnout before the candidate ran for office from the percent voter turnout from after the election. Assume that the populations are normally distributed. The firm uses the alternative hypothesis Ha:μd>0. Suppose the test statistic t is computed as t≈5.41, which has 9 degrees of freedom. What is the p-value for the hypothesis test, rounded to three decimal places? P Value =arrow_forward.A sample of 9 measurements, randomly selected from a normally distributed population, resulted in x= 2.6, and s= 0.9 Conduct a hypothesis test to verify the claim that the population mean is greater than 2.5 . Use a=.05arrow_forwardA hypothesis test produces a t statistic of t=2.01.If the researcher is using a two tailed test with x=0.05,how large does the sample have to be in order to reject the null hypothesis?arrow_forward
- Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…arrow_forwardConsider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)arrow_forwardFor 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield x=6.4, y=6.0, r=−0.143, P-value=0.322, and y=7.15−0.174x. Find the best-predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=7. Use a 0.05 significance level.arrow_forward
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