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Chapter 5: Q15E (page 314)

Will the sampling distribution of x炉always be approximately normally distributed? Explain

Short Answer

Expert verified

Answer

A sampling distribution is statistics derived by continuous sampling from a greater populace.

Step by step solution

01

Step-by-Step Solution Step 1: Sampling distribution

A sampling distributionis a probabilistic distribution of a statistic resulting from the selection of randomized samples from a particular population. It reflects the distribution of frequency on how far apart certain events will be for a specific demographic.

02

Explanation

No, since the central limit theorem asserts that if the sample size increases sufficient, the sampling distributions of x overbar are nearly distributed normally.

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Most popular questions from this chapter

Refer to Exercise 5.3 and find E=(x)=渭. Then use the sampling distribution ofxfound in Exercise 5.3 to find the expected value ofx. Note thatE=(x)=渭.

Variable life insurance return rates. Refer to the International Journal of Statistical Distributions (Vol. 1, 2015) study of a variable life insurance policy, Exercise 4.97 (p. 262). Recall that a ratio (x) of the rates of return on the investment for two consecutive years was shown to have a normal distribution, with 渭=1.5, 蟽=0.2. Consider a random sample of 100 variable life insurance policies and letx炉represent the mean ratio for the sample.

a. Find E(x) and interpret its value.

b. Find Var(x).

c. Describe the shape of the sampling distribution ofx炉.

d. Find the z-score for the value x炉=1.52.

e. Find Px炉>1.52

f. Would your answers to parts a鈥揺 change if the rates (x) of return on the investment for two consecutive years was not normally distributed? Explain.

Question: The standard deviation (or, as it is usually called, the standard error) of the sampling distribution for the sample mean, x炉 , is equal to the standard deviation of the population from which the sample was selected, divided by the square root of the sample size. That is

蟽X炉=蟽n

  1. As the sample size is increased, what happens to the standard error of? Why is this property considered important?
  2. Suppose a sample statistic has a standard error that is not a function of the sample size. In other words, the standard error remains constant as n changes. What would this imply about the statistic as an estimator of a population parameter?
  3. Suppose another unbiased estimator (call it A) of the population mean is a sample statistic with a standard error equal to

蟽A=蟽n3

Which of the sample statistics,x炉or A, is preferable as an estimator of the population mean? Why?

  1. Suppose that the population standard deviation 蟽is equal to 10 and that the sample size is 64. Calculate the standard errors of x炉and A. Assuming that the sampling distribution of A is approximately normal, interpret the standard errors. Why is the assumption of (approximate) normality unnecessary for the sampling distribution ofx炉?

Consider the following probability distribution:

a. Calculate for this distribution.

b. Find the sampling distribution of the sample mean for a random sample of n = 3 measurements from this distribution, and show that is an unbiased estimator of .

c. Find the sampling distribution of the sample median for a random sample of n = 3 measurements from this distribution, and show that the median is a biased estimator of .

d. If you wanted to use a sample of three measurements from this population to estimate , which estimator would you use? Why?

Suppose a random sample of n = 25 measurements are selected from a population with mean 渭and standard deviation s. For each of the following values of 渭and role="math" localid="1651468116840" 蟽, give the values of 渭蠂炉 and 蟽蠂炉.

  1. 渭=100,蟽=3
  2. 渭=100,蟽=25
  3. 渭=20,蟽=40
  4. 渭=10,蟽=100
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