Lectures 9 & 10 Conceptual Short-Answer

Question 1

What is the sampling distribution?

Answer 1

Refers to the probability distribution of a sample statistic for a given sample size (N)

Question 2

What is the benefit of using the sampling distribution?

Answer 2

Allows us to make statistical inferences about population parameters using a sample

Question 3

What is the random sampling?

Answer 3

Refers to a process of selecting a subset of cases from a larger population at random

Question 4

When can we call a sample statistic an estimator?

Answer 4

The sample statistics that are used to estimate population parameters are called estimators

Question 5

What is the difference between estimators and estimates?

Answer 5

Estimators refer to the sample statistics used to estimate population parameters and estimates refer to the specific values of the estimator

Question 6

When can a random samling be called a simple random sampling?

Answer 6

When each case has an equal chance of being selected

Question 7

Describe two advantages of random sampling.

Answer 7

1) increases likelihood that the sample is representative of the population
2) allows us to establish the probability distribution of a sample statistic

Question 8

When we collect a sample using random sampling, we can treat a sample statistic as a random variable. Explain why.

Answer 8

Since random sampling = random experiment we can treat a sample statistic as a random variable

Question 9

Let 𝜇 and 𝜎 denote the population mean and standard deviation, respectively, of a random variable X.
What is the Central Limit Theorem (CLT)?

Answer 9

CLT means that when a sample is collected by random sampling and N is sufficiently large (e.g., 𝑁≥
30), X̅ approximately follows 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇,𝜎/√𝑁), regardless of the population distribution.

Question 10

What is the difference between point and interval estimators?

Answer 10

Point estimator refers to a sample statistic used to estimate the value of a parameter as a single point.

Interval estimator refers to an interval that may contain the population parameter with a certain probability.

Question 11

When we calculate a sample statistic using data in our sample and employ it to estimate the target parameter, an error may occur. When a sample is randomly collected, there could be 2 sources of error. Explain each of them in a single sentence.

Answer 11

Bias: a systematic error that occurs due to an inherent property of the estimator
Sampling error: a random error that occurs as the parameter are estimated based on a sample

Question 12

An estimator is said to be unbiased if 𝐸(θ̂ ) = θ. Explain what it means.

Answer 12

If we can repeat random sampling and obtain multiple sample statistic values, their average will be close to the population parameter θ

Question 13

An estimator is said to be biased if 𝐸(θ̂ ) ≠ θ. Explain what it means.

Answer 13

Even if we repeat random sampling and obtain an infinite number of sample statistic values, their average will be different from the population parameter θ

Question 14

The standard deviation of the sampling distribution for an unbiased estimator is called a standard error (SE). What does SE represent?

Answer 14

The expected amount of error when θ̂ is used to estimate θ

Question 15

95% confidence interval (CI) of a sample mean refers to an interval that may include 𝜇 with 95% probability. What does this 95% probability mean?

Answer 15

When one repeats random sampling and constructs 95% CIs multiple times, 95% of them are likely to include the true population mean

Question 16

What does α mean in the 100(1- α)% confidence interval (CI) of a sample mean?

Answer 16

It refers to an error rate (or risk) accepted by researchers that the confidence interval may not capture the true population mean