Applied Economics & Statistics: Topic 5 - Estimation and Confidence Intervals*

Question 1

Why do we estimate?

Answer 1

• We usually don’t have access to data for the whole population.
• Inferential Statistics estimates population parameters from sample statistics.

Example: Determine the mean income of UK residents.
We don’t have access to everyone’s income values, and it’s too costly
and time consuming to collect this information.
Instead, take a random representative sample of UK residents (using
an appropriate sampling technique).
Use an estimator, the sample mean, to estimate the population value
for mean income: a point estimate

Question 2

Describe what makes a good estimator in statistics

Answer 2

• We want our estimator to be accurate.
• If we don’t know the population parameter, how do we know the
  estimator is close to it?
• Estimator must have “good properties.”
  Specifically, it must be unbiased and efficient.

Question 3

When is an estimator bias?

Answer 3

An estimator is biased if the mean of its sampling distribution is not
equal to the population value of interest.

Question 4

Explain whether the ‘sample mean’ is bias or not

Answer 4

We know from the Central Limit Theorem that the sample mean is an
unbiased estimator of the population mean, because its sampling
distribution is centred on the population mean

Question 5

When choosing between multiple unbiased estimators, which one is preferred?

Answer 5

The one with
least variance (more efficient) is preferred.

Question 6

Describe how we can tell when an estimator has a high efficiency

Answer 6

An estimator with high efficiency will have a low standard deviation in
its sampling distribution.
So a high degree of clustering of values

Question 7

What does it mean when an estimator has low standard deviation?

Answer 7

It has a high efficiency

Question 8

Describe how the efficiency of estimators can be improved

Answer 8

• We can improve the efficiency of estimators by increasing the size of
  the sample used.
• Recall the standard deviation of the sample mean is σ/√n.
• So as n increases, the standard deviation gets smaller

Question 9

What does increasing sample size do to an estimator?

Answer 9

It increases its efficiency

Question 10

What does an unbiased and efficient estimator do?

Answer 10

It gives us the best chance of
getting an estimate close to the true value

Question 11

Which estimator gives us the best chance of
getting an estimate close to the true value?

Answer 11

Unbiased and efficient estimator

Question 12

What does a biased but efficient estimator do?

Answer 12

It would provide estimates that are
clustered but around the wrong value

Question 13

Which estimator would provides estimates that are
clustered but around the wrong value?

Answer 13

A biased but efficient estimator

Question 14

What does a biased and inefficient estimator do?

Answer 14

We wouldn’t even get close on
average, let alone with a single sample.

Question 15

What type of estimator wouldn’t even get close on
average, let alone with a single sample?

Answer 15

A biased and inefficient estimator

Question 16

In practice, describe which estimators we use

Answer 16

In practice, we might be willing to use an estimator that is biased but the most efficient, if its bias is small. But of course we always prefer
to use the most efficient (minimum variance) and unbiased (on average correct) estimator

Question 17

What are ‘estimates’?

Answer 17

Values derived from a sample and used to approximate
population values

Question 18

What’s a ‘point estimate’?

Answer 18

The statistic, computed from sample information, that estimates a
population parameter. Example: the maximum temperature tomorrow will be 15C. We can think
of this as our “best guess” as to what tomorrow’s temperature will be

Question 19

What’s the name for ‘the statistic, computed from sample information, that estimates a
population parameter’?

Answer 19

Point Estimate

Question 20

What are ‘confidence intervals’?

Answer 20

A range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the confidence level. Example: the maximum temperature tomorrow will be between 13C and
17C

Question 21

What’s the name for ‘a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability’?

Answer 21

Confidence Intervals

Question 22

What are confidence levels often stated as?

Answer 22

• Confidence intervals are often stated as CI = point estimate ± margin of
  error:
  1. We find a point estimate by using the sample mean, ̄x , as we have
  already seen in previous topics.
  2. But we also state this as a CI for which we provide a range of
  values and a measure of how certain we are that this range contains
  the true population mean value.
  Example: For example: “At a confidence level of 95%, between 52% and
  58% of Americans favour the death penalty for people convicted of
  murder.” i.e. “The support for death penalty is currently 55%(±3%).”

Question 23

What are the 2 possible situations for calculating confidence intervals for a population mean?

Answer 23

1. We use sample data to estimate μ with ̄x and the population standard deviation σ is known.
2. We use sample data to estimate μ with ̄x and the population standard deviation σ is unknown. In this case, we use sample standard deviation s

Question 24

Describe & explain how we find CIs when σ is known

Answer 24

• The distribution of the sample mean is:
  ̄x ∼ N (μ, σ^2/n),
  and we know that
  z = ( ̄x − μ) / (σ/√n) ∼ N(0, 1).
• We use this information to determine probabilities associated with ̄x
  taking values in certain ranges, and from that build CIs
• We often want to find a 95% confidence interval, i.e. one in which we are
  95% confident the true population mean lies.
  For what values of z would there be 95% of the area under the standard
  normal distribution curve? You can read this from a z table and you
  should find that the value is 1.96. So, 95% of the area lies in the interval
  (−1.96, +1.96), or:…