Chapter 7 (Statistical Estimation and Sampling Distributions)

Question 1

What is a parameter?

Answer 1

Numeric characteristic of the population. Denoted with theta

Question 2

What is a statistic?

Answer 2

A numeric characteristic of a sample used to estimate the population parameter. Denoted theta_hat

Question 3

What is a point estimate?

Answer 3

The realized value of a statistic.

Question 4

What is a sampling distribution?

Answer 4

Consists of all possible values of a statistic for a given sample size, and their associated probabilities.

Question 5

What can a statistic be seen as?

Answer 5

A random variable!

Question 6

What characteristics of the statistic can we use to ascertain how good the statistic is as an estimator of the parameter?

Answer 6

The sampling distribution, with its expected value and variance.

Question 7

What is the Central Limit Theorem?

Answer 7

If we have a sample (iid) from a distribution with myu and sigma^2, then if the sample n is large enough, the distribution is approximately normal:

X_bar •~ N(µ, sigma^2/n)
nX •~ N(nµ, nsigma^s)
p_hat •~ N(p, [p(1-p)]/n)

Question 8

Using the CLT, what is the sample proportion distribution?

Answer 8

p_hat •~ N(p, p(1-p)/n)

Question 9

How do we know if a statistic is unbiased?

Answer 9

If the expected value of the statistic is equal to the parameter.

E(theta_hat) = theta
E(µ_hat) = µ
E(sigma_hat) = sigma

Question 10

If a statistic is biased, what is the bias?

Answer 10

The bias is:

E(theta_hat) - theta.

Question 11

How do we compute the population variance?

Answer 11

Sigma^2 = 1/n(Sum(xi- myu)^2

Question 12

How do we compute the sample variance? (S^2)

Answer 12

S^2 = 1/(n-1)(Sum(Xi- X_bar)^2

Question 13

What does the variance of a statistic tell us?

Answer 13

How much the value of the statistic is likely to change from sample to sample.

Question 14

What is the standard deviation of a statistic called?

Answer 14

Standard error.

Question 15

What is the formula for standard error?

Answer 15

s/sqrt(n)

Question 16

We want our statistics to be:

Answer 16

Unbiased, and with low variance.

Question 17

What is the margin of error?

Answer 17

What percentage of a sample means we expect to be within a certain distance from a parameter.

Specified from the sampling distribution.

Question 18

Why do we use the t-distribution?

Answer 18

Because if our sample size is small, it’s fairly accurate, and if its large it’s VERY accurate!

Question 19

If we don’t know the population variance, then we have to estimate it as well as the mean, which means we get this formula and this distribution:

Answer 19

Sqrt(n)*(X_bar - µ)
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s

~ t_{n-1}