Concept explainers
An estimator θ is said to be consistent if for any ∈ > 0,
Now identify Y with
Show that
Explanation of Solution
Calculation:
Chebyshev’s inequality can be rewritten as:
The random variable considered here is the sample mean,
The quantity
Replace Y by
When
As a result, when
Thus, using Chebyshev’s inequality, it can be shown that
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Chapter 6 Solutions
DEVORE'S PROB & STATS F/ENG WEBASSIGN
- Suppose X has hypergeometric distribution with parameter N=1000, M=150, and n=10. Then the set { X ≤ 3 } can be interpreted as Selectallthatapply: the set of all samples of size 10 containing less than or equal to 3 successes, when sampled with replacement from a population of size 1000 and containing 150 successes the set of all samples of size 10 containing less than or equal to 3 successes, when sampled with without replacement from a population of size 150 and containing 10 successes.. the set of all samples of size 10 containing less than or equal to 3 successes, when sampled with without replacement from a population of size 1000 and containing 150 successes. . the set of all samples of size 10 containing more than 3 successes, when sampled with without replacement from a population of size 1000 containing 150 successes.arrow_forwardIf 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.arrow_forwardIf X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.arrow_forward
- A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.arrow_forwardA poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?arrow_forwardA researcher is using a two-tailed hypothesis test with α = 0.01 to evaluate the effect of a treatment. If theboundaries for the critical region are t = ± 2.845, then how many individuals are in the samplearrow_forward
- A producer of pocket calculators purchases the main processor chips in lots of1,000. The producer would like to have a 1 percent rate of defectives but willnormally not refuse a lot unless it has 4 percent or more defectives. Samples of50 are drawn from each lot, and the lot is rejected if more than two defectives arefound.b. Compute A and B Use the Poisson approximation for your calculationsarrow_forwardA 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_forwardA student attempted to use L’Hopital’s Rule as follows. Did the student make an error, if any, or does it state “no error”?arrow_forward
- A researcher is using a two-tailed hypothesis test with α = 0.05 to evaluate the effect of a treatment. If the boundaries for the critical region are t = ± 2.080, then how many individuals are in the sample?arrow_forwardIf X1, X2, and X3 constitute a random sample of sizen = 3 from a Bernoulli population, show that Y =X1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint:Consider special values of X1, X2, and X3.)arrow_forwardA 99 percent one-sample z-interval for a proportion will be created from the point estimate obtained from each oftwo random samples selected from the same population: sample R and sample S. Let R represent a random sampleof size 1,000, and let S represent a random sample of size 4,000. If the point estimate obtained from R is equal tothe point estimate obtained from S, which of the following must be true about the respective margins of errorconstructed from those samples?(A) The margin of error for S will be 4 times the margin of error for R.(B) The margin of error for S will be 2 times the margin of error for R.(C) The margin of error for S will be equal to the margin of error for R.(D) The margin of error for R will be 4 times the margin of error for S.(E) The margin of error for R will be 2 times the margin of error for S.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning