leiyou

 

2、If the variance of the sampling distribution of an estimator is smaller than all other unbiased estimators of the parameter of interest, the estimator is:

A) efficient.

B) reliable.

C) unbiased.

D) consistent.

leiyou

 

The correct answer is A

By definition.

leiyou

 

3、If Estimator A is a more efficient estimator than Estimator B, it will have:

A) a smaller mean and the same variance. 

B) the same mean and a smaller variance. 

C) a smaller mean and a larger variance. 

D) the same mean and a larger variance.

leiyou

 

The correct answer is B

The more efficient estimator is the one that has the smaller variance, given that they both have the same mean.

leiyou

 

AIM 9: Define Best Linear Unbiased Estimator.

1、Shawn Choate is thinking about his graduate thesis. Still in the preliminary stage, he wants to choose a variable of study that has the most desirable statistical properties. The statistic he is presently considering has the following characteristics:

?            The expected value of the sample mean is equal to the population mean.

?            The variance of the sampling distribution is smaller than that for other estimators of the parameter.

?            As the sample size increases, the standard error of the sample mean rises and the sampling distribution is centered more closely on the mean.

?            Select the best choice. Choate’s estimator is:

A) unbiased, efficient, and consistent.

B) efficient and consistent.

C) unbiased and consistent.

D) unbiased and efficient.

leiyou

 

The correct answer is D

The estimator is unbiased because the expected value of the sample mean is equal to the population mean. The estimator is efficient because the variance of the sampling distribution is smaller than that for other estimators of the parameter.

The estimator is not consistent. To be consistent, as the sample size increases, the standard error of the sample mean should fall and the sampling distribution will be centered more closely on the mean. A consistent estimator provides a more accurate estimate of the parameter as the sample size increases.

leiyou

 

AIM 5: Define, calculate and interpret confidence interval, confidence coefficient, upper limit, lower limit, random interval.

1、The average U.S. dollar/Euro exchange rate from a sample of 36 monthly observations is $1.00/Euro. The population variance is 0.49. What is the 95% confidence interval for the mean U.S. dollar/Euro exchange rate?

A) $0.7713 to $1.2287.

B) $0.5100 to $1.4900.

C) $0.8075 to $1.1925.

D) $0.8657 to $1.1343.

leiyou

 

The correct answer is A

The population standard deviation is the square root of the variance (√0.49 = 0.7). Because we know the population standard deviation, we use the z-statistic. The z-statistic reliability factor for a 95% confidence interval is 1.960. The confidence interval is $1.00 ± 1.960($0.7 / √36) or $1.00 ± $0.2287.

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2、A sample of size 25 is selected from a normal population. This sample has a mean of 15 and the population variance is 4.

Using this information, construct a 95% confidence interval for the population mean, m.

A) 15 ± 1.96(0.4).

B) 15 ± 1.96(2).

C) 15 ± 1.96(0.8).

D) 15 ± 1.96(4).

leiyou

 

The correct answer is A

Because we can compute the population standard deviation, we use the z-statistic.  A 95% confidence level is constructed by taking the population mean and adding and subtracting the product of the z-statistic reliability (zα/2) factor times the known standard deviation of the population divided by the square root of the sample size (note that the population variance is given and its positive square root is the standard deviation of the population): x ± zα/2 × ( σ / n1/2) = 15 ± 1.96 × (41/2 / 251/2) = 15 ± 1.96 × (0.4).