Topic 7 - Estimation

Question 1

What is an estimator

Answer 1

• An estimator of a population parameter is a random variable that depends on sample information

Question 2

What does an estimator provide

Answer 2

• An approximation to an unkown parameter

Question 3

What is an estimate

Answer 3

• A specific value of the estimator random variable

Question 4

What are point and interval estimates

Answer 4

• A point estimate is a single number
• A confidence interval provides additional information about variablility

Question 5

What is the first property of an estimator

Answer 5

• Unbiasedness
• A point estimator θ hat is said to be an unbiased estimator of the parameter θ if the expected value, or mean of the sampling distribution of θ hat is θ
• E(θ hat) = θ

Question 6

What does it mean if we have a biased estimator

Answer 6

• The estimate we see comes from a distribution which is not centered around the real parameter

Question 7

How is the Bias of an estimator calculated

Answer 7

• Bias(θ hat) = E(θ hat) - θ
• The bias of an unbiased estimator is 0

Question 8

What is the second property of an estimator

Answer 8

• Efficiency
• If we have multiple unbiased estimators of θ, the most efficient is the one with the smallest variance
• If var(θ hat 1) < var(θ hat 2) then θ hat 1 is more efficient

Question 9

How is the relative efficiency of two unbiased estimators calculated

Answer 9

• var(θ hat 2) / var(θ hat 1)
• Relative efficiency of theta hat 1 with respect to theta hat 2

Question 10

What do both properties consistency and unbiasedness assume

Answer 10

• That the data consists of a fixed sample size n

Question 11

What is the third property of an estimator

Answer 11

• Consistency
• Consistency is an asymptotic property, concerning the study of behaviour of an estimator as the sample size increases indefinetly
• Concerns how far the estimator is likely to be from the parameter it is estimating as the sample increases

Question 12

When is an unbiased estimator always consistent

Answer 12

• If variance shrinks to 0 as the sample size grows

Question 13

What is a confidence interval estimator

Answer 13

• A rule of determining an interval that is likely to include the parameter

Question 14

• How are confidence intervals calculated

Answer 14

• If P(a < θ < b) = 1 - alpha then the interval from a to b is called a 100(1-alpha)% confidence interval of θ
• 100(1-alpha)% is called the confidence level of the interval (0 < alpha < 1)
• This is written as a < θ < b with 100(1-alpha)% confidence

Question 15

If σ^2 is known, what is the estimate for our confidence interval

Answer 15

• x bar - z(alpha/2) * σ/sqrt(n) < mu < x bar + z(alpha/2) * σ/sqrt(n)
• where z(alpha/2) is the normal distribution value for a probability for alpha/2 in each tail

Question 16

What is the margin of error and how can the confidence interval estimate when σ^2 is known be rewritten

Answer 16

• ME = z(alpha/2) * σ/sqrt(n)
• x bar - ME < mu < x bar + ME

Question 17

How large is the interval width

Answer 17

• 2 * Margin of Error

Question 18

How can the margin of error be reduced

Answer 18

• Population S.D reduced
• Sample size increases
• Confidence level (1 - alpha) decreases

Question 19

What do we do if the population standard deviation is unkown

Answer 19

• We can substitute the sample standard deviation (s)

Question 20

What does substiuting s for σ cause

Answer 20

• Greater uncertainty, as s varies from sample to sample

Question 21

How is the Students t variable calculated when σ is unkown

Answer 21

• t = x bar - mu / (s / sqrt(n))
• With (n - 1) degrees of freedom

Question 22

What happens with the t distribution as n increases

Answer 22

• t tends to Z

Question 23

What is the confidence interval for mu if σ is unkown

Answer 23

• x bar - t(n-1,a/2) * S/sqrt(n) < mu < x bar + t(n-1,a/2) * S/sqrt(n)
• Where t(n-1,a/2) is the critical value of the t ditribution with n-1 degrees of freedom and an area of a/2 in each tail