Econ Stats Module 8+9

Population

The entire group of individuals or items we want information about.

Sample

A subset of the population used to draw conclusions about the population.

Research

The process of studying a problem, question, or hypothesis by collecting and analyzing data.

Primary goal of sampling

To obtain an unbiased sample that is representative of the population.

Convenience sample

A sample chosen because it is easy to obtain; it often produces biased results.

Simple random sample

A sample in which every member of the population has the same chance of being selected.

Systematic sample

A sample taken by choosing a random starting point and then selecting every kth item.

Stratified sample

A sample taken by dividing the population into strata and randomly sampling from each stratum.

Cluster sampling

A sample taken by dividing the population into primary units or clusters and then sampling those units.

Reasons to sample instead of study the whole population

Sampling saves time, reduces cost, may be physically necessary, and avoids destructive testing of every item.

Sampling error

The difference between a sample statistic and its corresponding population parameter.

Sample statistic

A numerical value computed from a sample, such as a sample mean or sample proportion.

Population parameter

A numerical value that describes a population, such as μ, σ, or π.

Sampling distribution of the sample mean

The probability distribution of all possible sample means for a given sample size.

Central Limit Theorem (CLT)

If samples of size n are taken from a population, the sampling distribution of the sample mean is approximately normal for sufficiently large n.

CLT for a normal population

If the population itself is normal, the sampling distribution of the sample mean is normal for any sample size.

When does the sample mean become more normal?

As sample size increases, the sampling distribution becomes more nearly normal.

Rule of thumb for CLT with a symmetric population

If the population is symmetric, a sample size as small as 10 may be enough for the sample mean to look normal.

Rule of thumb for CLT with a skewed population

If the population is skewed or has thick tails, a sample size of 30 or more may be needed.

Mean of the sampling distribution of x-bar

The mean of the sample means equals the population mean: μx̄ = μ.

Standard error of the mean

The standard deviation of the sampling distribution of the sample mean: σx̄ = σ/√n.

Effect of increasing sample size on standard error

As n increases, the standard error decreases.

Effect of increasing sample size on the sampling distribution

The distribution gets narrower, less dispersed, and more normal in shape.

z-score for a sample mean

z = (x̄ - μ) / (σ/√n)

Condition for using the sampling distribution of x-bar

Use it when the population is normal or when n is at least 30 and the population shape is unknown.

Proportion

The fraction, ratio, or percent of the sample or population with a trait of interest.

Sample proportion formula

p = x/n

Population proportion symbol

π

Binomial conditions for sample proportion

Two outcomes, independent trials, and constant probability of success.

Conditions for normal approximation for p

nπ ≥ 5 and n(1 - π) ≥ 5

Mean of the sampling distribution of p

The mean of the sample proportion distribution is π.

Standard error of the sample proportion

σp = √[π(1 - π)/n]

z-score for a sample proportion

z = (p - π) / √[π(1 - π)/n]

Point estimate

A single sample statistic used to estimate a population parameter.

Point estimate for population mean

x̄ is the point estimate of μ.

Confidence interval

A range of values likely to contain a population parameter at a specified confidence level.

Confidence level

The percentage of similarly constructed intervals that would contain the true population parameter.

Why use a confidence interval?

It accounts for sampling error and gives a range of plausible values for the population parameter.

Confidence interval for μ when σ is known

x̄ ± z(σ/√n)

95% z-value

1.96

90% z-value

1.65

Margin of error for μ when σ is known

E = z(σ/√n)

What determines the width of a confidence interval?

The confidence level and the standard error.

What affects the standard error for a mean?

The population standard deviation σ and the sample size n.

Confidence interval for μ when σ is unknown

x̄ ± t(s/√n)

Degrees of freedom for a t-interval

df = n - 1

Why use the t-distribution instead of z?

Use t when the population standard deviation σ is unknown and is estimated with the sample standard deviation s.

Shape of the t-distribution

It is symmetric and mound-shaped, but flatter and more spread out than the standard normal distribution.

How does the t-distribution change as df increases?

As degrees of freedom increase, the t-distribution approaches the z-distribution.

Why are t critical values larger than z critical values?

Because the t-distribution has more spread, so a larger cutoff is needed for the same confidence level.

Assumption for using the t-interval

The population is assumed to be normal, especially for small samples.

Confidence interval for a population proportion

p ± z√[p(1 - p)/n]

Standard error of a sample proportion using sample data

√[p(1 - p)/n]

Margin of error for a population proportion

E = z√[p(1 - p)/n]

Finite population correction factor

√[(N - n)/(N - 1)]

When use the finite population correction

When sampling from a relatively small finite population, it reduces the standard error.

Effect of the finite population correction

It makes the estimate more precise by reducing the standard error.

Choosing sample size for estimating a mean

Sample size depends on margin of error, confidence level, and variation.

Margin of error symbol

E

Margin of error for estimating a mean

E = zσ/√n

Required sample size for estimating a mean

n = (zσ/E)^2

How to handle non-whole-number sample size for a mean

Round up to the next whole number.

Margin of error for estimating a proportion

E = z√[π(1 - π)/n]

Required sample size for estimating a proportion

n = π(1 - π)(z/E)^2

What value of π should be used if no estimate is available?

Use π = 0.50.

Why use π = 0.50 when no estimate is available?

Because π(1 - π) is largest at 0.50, giving the most conservative, largest required sample size.

Interpretation of a 95% confidence interval

About 95% of intervals built this way from repeated random samples would contain the true population parameter.

Sampling distribution vs population distribution

A population distribution describes individual values; a sampling distribution describes a statistic computed from many samples.

Difference between standard deviation and standard error

Standard deviation describes spread in individual data; standard error describes spread in a sampling distribution.

Formula card: sample proportion

p = x/n

Formula card: mean of sampling distribution of x-bar

μx̄ = μ

Formula card: standard error of x-bar

σx̄ = σ/√n

Formula card: z-score for x-bar

z = (x̄ - μ) / (σ/√n)

Formula card: standard error of p

σp = √[π(1 - π)/n]

Formula card: z-score for p

z = (p - π) / √[π(1 - π)/n]

Formula card: confidence interval for μ, σ known

x̄ ± z(σ/√n)

Formula card: confidence interval for μ, σ unknown

x̄ ± t(s/√n)

Formula card: degrees of freedom

df = n - 1

Formula card: confidence interval for π

p ± z√[p(1 - p)/n]

Formula card: finite population correction

√[(N - n)/(N - 1)]

Formula card: sample size for mean

n = (zσ/E)^2

Formula card: sample size for proportion

n = π(1 - π)(z/E)^2