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EBK AN INTRODUCTION TO MATHEMATICAL STA
- The distance between large cracks in a road follows a Poisson Process with an average of 3 cracks per kilometer. What is the probability that at most 1/2 kilometer will have to be covered before the next large crack appears? [Note that distance is a continuous variable similar to time]arrow_forwardA sample of 120 employees of a company is selected, and the average age is found to be 37 years. Parameter or stasistic?arrow_forwardThe number of visits to a website is known to have a Poisson distribution with a mean of 10 visits per minute Within what limits does Tchebysheff's Theorem suggest you would expect the number of visits to this website to lie at least 75% of the time?arrow_forward
- Suppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha=50$ and $\beta=2 .$ Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.arrow_forwardSuppose that the makers of Duracell batteries want to demonstrate that their size AAbattery lasts an average of at least 45 minutes longer than Duracell’s main competitor,the Energizer. Two independent random samples of 100 batteries of each kind areselected, and the batteries are run continuously until they are no longer operational. Thesample average life for Duracell is found to be 308 minutes. The result for the Energizerbatteries is 254 minutes. Assume the entire AA batteries standard deviation 84 minutesand Duracell batteries 67 minutes. Is there evidence to substantiate Duracell’s claim thatits batteries last, on average, at least 45 minutes longer than Energizer batteries of thesame size? (assuming equal population variances)arrow_forwardLet X1, X2, …, Xn be a random sample from the Normal distribution N() (a) Using method of moments to estimate the parameters and . (b) Are those estimators unbiased?arrow_forward
- If X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.arrow_forwardIf X is a continuous random variable, what is the value of X that will be exceeded 30% of the time, if X has a gamma distribution with a mean of 2 and a variance of 2? (just set up the equation in itssimplest form)arrow_forwardAssuming alpha level is held constant, if someone does a one-tailed test the critical value will be____ than if he or she did a two tailed test.arrow_forward
- A simple random sample of size n =66, is obtained from a population that is skewed left with =33 and =3. . Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? What is the sampling distribution of x? Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why?(A) Yes. The central limit theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. (B) Yes. The central limit theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x normal, regardless of the sample size, n. (C)No. The central limit theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x, become approximately normal as the sample size, n, increases. (D) No. The central limit theorem…arrow_forwardAn insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is λ is Poisson distributed with mean λ. They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters s and α. (a) If a newly insured person has n accidents in her first year, find the conditional density of her accident parameter. (b) Also, determine the expected number of accidents that she will have in the following year.arrow_forwardSuppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ = 20 (suggested in article "Dynamic Ride Sharing: Theory and Practice," J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will: (i) Be at most 10? (ii) Exceed 20? (iii) Be between 10 and 20, inclusive? (iv) Be strictly between 10 and 20?arrow_forward
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