Lecture 8; Sampling Distributions:

Question 1

What is one primary goal of statistics?

Answer 1

One primary goal of statistics is to estimate (infer) an unknown quantity (parameter) of a population based on sample data.

Question 2

What does estimation involve?

Answer 2

Estimation involves inferring a population parameter (e.g., mean, standard deviation, median) from a sample.

Question 3

What is variation within a sample summarized by?

Answer 3

The variation within a sample is summarized by the standard deviation

Question 4

What does variation within a sample tell us?

Answer 4

The variation within a sample, summarized by the standard deviation, informs us about the expected variability of sample means around the true population mean—that is, how much our sample average might deviate simply because of random sampling.

Question 5

What is Statistical “superpower”?

Answer 5

By measuring variation within a sample, we can estimate how uncertain our estimate is—that is, how much the sample means would fluctuate if we repeated the study.

Question 6

What is a population parameter vs a sample estimate and provide an example?

Answer 6

• A parameter describes a quantity in a statistical population, while an estimate (or statistic) is a similar quantity derived from a sample.
• For example, the mean of a population is a parameter, whereas the mean of a sample is an estimate (or statistic) of the population mean.

Question 7

Why is an an estimate (derived from a sample) rarely, if ever, exactly the same as the population parameter being estimated—especially in large populations?

Answer 7

because sampling is influenced by chance.

Question 8

At its core, statistics asks…

Answer 8

At its core, statistics asks: given uncertainty arising from random sampling, how reliable is an estimate, and how much confidence should we place in decisions based on it? Put differently, how close is a sample-based estimate expected to be to the true population value?

Question 9

Is the goal of statistics to quantify or eliminate uncertainty?

Answer 9

The goal is not to eliminate uncertainty (one can’t), but to quantify it, so that we can make decisions with a known degree of certainty.

Question 10

What are 3 examples of properties of estimators?

Answer 10

1) Mean
2) Variance
3) Sd

Question 11

What is a sampling distribution and what does it describe?

Answer 11

• A sampling distribution is the probability distribution of an estimator generated by random sampling from a population.
• It describes what values the estimate would take if we were to repeatedly sample from the same population under identical conditions.

Question 12

How do sampling distributions and frequency distributions differ?

Answer 12

Sampling distributions describe probabilities of possible estimates, not frequencies of observed data values

Question 13

What is µ?

Answer 13

µ = population mean (we say “mu”, Greek alphabet).

Question 14

What is σ?

Answer 14

σ = population standard deviation (we say “sigma”)

Question 15

What is σ^2?

Answer 15

σ2 = population variance (we say “sigma squared”)

Question 16

What is X̄ or Ȳ?

Answer 16

X̄ = sample mean (we say “X bar”, LaBn or Roman alphabet).
–> While µ always represents the population mean of a variable (e.g., X), the symbol for the sample mean depends on the variable being measured. For example, the sample mean of X is written as X̄, and the sample mean of Y is written as Ȳ. The key point is that the sample mean is always indicated by a bar over

Question 17

What is s?

Answer 17

s = sample standard deviation

Question 18

What is s^2?

Answer 18

sample variance

Question 19

What is the first property of sampling distributions?

Answer 19

Property 1: The mean of all sample means is always equal to the population mean

Question 20

When the mean of all possible sample means = the population parameter, the estimate is said to be…

Answer 20

the sample mean is unbiased because, under random sampling, the sample means do not systematically tend to be either larger or smaller than the true population mean

Question 21

When sampling is biased, what does this mean about the differences in sample estimate and population value?

Answer 21

When sampling is biased, differences between the sample estimate and the population value are no longer due solely to random variation—they reflect a systematic distortion of the sampling process

Question 22

How does sample size affect the precision of sample estimates under random sampling?

Answer 22

• As sample size increases, sampling variability decreases, so sample means cluster more tightly around the true population mean, yielding more precise estimates.

Question 23

Does random sampling eliminate sampling error?

Answer 23

Random sampling does not eliminate sampling error, but it prevents systematic bias and allows sampling error to be measured

Question 24

What is the 2nd property of sampling distributions?

Answer 24

Property #2: Under random sampling, increasing sample size reduces sampling variability, so sample means cluster more tightly around the true population mean, resulting in greater precision.

Question 25

What are sampling distributions best represented by when the estimator is continuous?

Answer 25

Sampling distributions are best represented by probability density functions (PDFs) when the estimator is continuous, as is the case for sample means except in trivial or extremely small populations

Question 26

What does probability density give you?

Answer 26

Probability density does not give the probability of a single exact value, but rather how densely probability is concentrated around that value. Probabilities are obtained by integrating the PDF over an interval.

Question 27

How is probability density affected by sample means (sample means near or far from the pop mean)?

Answer 27

- Sample means near the true population mean occur often → they have high probability density. - Sample means far from the population mean occur rarely → they have low probability density

Question 28

How do you plot the probability distribution of the sample mean?

Answer 28

If you plot all possible sample means and how frequently they occur, you get a smooth curve. That curve is the probability distribution of the sample mean

Question 29

What do the height of the curve and area under the curve of a probability distribution of the sample mean (curve) tell you?

Answer 29

- The height of the curve at a value tells you the probability density (how common nearby values are). - The area under the curve over an interval gives you an actual probability