Class 2: Foundations of Statistical Inference, Sampling Distribution, & Confidence Intervals

What are the reasons of sampling populations?

• impossibility

• cost

• adequacy

• time

What is a point estimator?

stat estimating parameter to provide point estimate

What is a point estimate?

best parameter value guess

What are some point estimator drawbacks?

• incorrect

• unknown proximity

• doesn’t reflect larger sample effect

What is an interval estimator or confidence interval?

stat estimating parameter to provide range estimate

What are the estimate influence factors?

• standard deviation

• sample size

• proportionate sample size

What is the sample size notation?

n

What is the relationship between sample size & estimate?

larger size = better estimate

What is the standard deviation notation?

σ

What is the relationship between variability & estimate?

lower variability = better estimate

What is the proportionate sample size formula?

\frac{n}{N}

What is the relationship between proportionate sample size & estimate?

larger proportion = better estimate

What is the reason for creating an interval around point estimate?

estimate quality

What is the standard deviation shortcut formula?

\sigma=\sqrt{\frac{\sum\left(x_{i}-\mu\right)^2}{N}}

SQUARE ROOT sum of (population point - population mean)^2 divided by population size

What is the method of developing a mean sampling distribution?

1. consider possible samples

2. create distribution table

3. distribute sample means

What is the mean sample distribution mean formula?

\mu_{\bar{x}}=\sum\bar{x}P\left(\bar{x}\right)

SUM sample mean * sample mean probability

What is the mean sample distribution variance formula?

\sigma^2=\sum\left(\bar{x}-\mu\right)^2P\left(\bar{x}\right)

SUM (sample point - sample mean)^2 * sample mean probability

What is the central limit theorem?

approximately normally distributed mean sampling distribution for 30+ sample sizes

What is the mean sample distribution formula?

\bar{X}\approx N\left(\mu,\frac{\sigma}{\sqrt{n}}\right)

normally distributed sample means (population mean, standard deviation divided by sample size square root)

What is the mean sample distribution z-value formula?

z=\frac{\left(\bar{x}-\mu\right)}{\frac{\sigma}{\sqrt{n}}}

DIVIDE sample mean - population mean by standard deviation divided by sample size square root

What formula is used to calculate a cumulative mean weight probability?

P\left(\bar{X}>a\right)=P\left(\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\right)

=P\left(Z>\ldots\right)

DIVIDE sample mean - population mean by standard deviation divided by sample size square root

What is the =NORM.DIST(x̄,μ,σ/√n,1) Excel function?

= NORMAL DISTRIBUTION (sample mean, population mean, standard deviation divided by sample size square root, cumulative)

sample mean ≤ test sample mean cumulative probability (left side)

What is the =NORM.S.DIST(z,1) Excel function?

= NORMAL STANDARDIZED DISTRIBUTION (z-value, cumulative)

sample mean z-value ≤ test sample mean z-value cumulative probability (left side)

What is the =1-NORM.DIST(x̄,μ,σ/√n,1) Excel function?

= 1 - NORMAL DISTRIBUTION (sample mean, population mean, square root of standard deviation devided by sample size, cumulative)

sample mean ≥ test sample mean cumulative probability (right side)

What is the =NORM.S.DIST(-z,1) Excel function?

= NORMAL STANDARDIZED DISTRIBUTION (negative z-value, cumulative)

sample mean z-value ≥ test sample mean z-value cumulative probability (right side)

What is the criteria for using a normal distribution to approximate a proportion sample distribution?

• np ≥ 5

• n(1-p) ≥ 5

What is a sample proportion?

population proportion success estimator

What is the sample proportion formula?

\hat{p}=\frac{X}{n}

DIVIDE success number by sample size

What is a sample proportion standard error?

p̂ standard deviation

What is the sample proportion standard error formula?

\sigma_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}

E\left(\hat{p}\right)=p

SQUARE ROOT expected value * (1 - expected value) divided by sample size

What is the sample proportion standardized normal distribution formula?

Z=\frac{\hat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}

DIVIDE proportion standard error - expected value by [expected value(1 - expected value) divided by sample size]

What comprises inference?

• estimate

• hypothesis test

What is the LCL formula?

\bar{X}-Z_{\frac{a}{2}\frac{\sigma}{\sqrt{n}}}

SUBTRACT sample mean & inverse confidence level /2 ‘s z-value * standard deviation divided by sample size square root

What is the UCL formula?

\bar{X}+Z_{\frac{a}{2}\frac{\sigma}{\sqrt{n}}}

ADD sample mean & inverse confidence level /2 ‘s z-value * standard deviation divided by sample size square root

What is a confidence level?

1-α probability statement that interval contains true mean

What is the half width interval formula?

Z_{\frac{a}{2}}\frac{\sigma}{\sqrt{n}}

MULTIPLY inverse confidence level /2 ‘s z-value & standard deviation divided by sample size square root

What is the full width interval formula?

2\left(Z_{\frac{a}{2}}\frac{\sigma}{\sqrt{n}}\right)

MULTIPLY inverse confidence level /2 ‘s z-value & standard deviation divided by sample size square root by 2

What is the relationship between interval width & info?

wider interval = littler info

What is the relationship between confidence level & interval width?

higher confidence level = wider confidence interval

• more confidence requires wider interval

What is the relationship between population standard deviation & interval width?

larger standard deviation = wider confidence interval

• more variable data requires wider interval

What is the relationship between sample size & interval width?

higher sample size = narrower confidence interval

• IF = confidence level

• BUT + data cost

What methods narrow confidence interval width?

• reduce confidence level

• increase sample size

What is the sample size interval formula?

\bar{x}\pm B

ADD & SUBTRACT sample mean & error margin

What is the error margin formula?

B=\frac{z_{\frac{a}{2}}\sigma}{\sqrt{n}}

DIVIDE inverse confidence level /2 ‘s z-value * standard deviation by n

What is the sample size formula knowing the z-value, confidence level, standard deviation, & error margin?

n=\left(\frac{z_{\frac{a}{2}}\sigma}{B}\right)^2

SQUARE inverse confidence level /2 ‘s z-value divided by error margin

• round up

What is the proportion confidence interval formula?

\hat{p}\pm z_{\frac{a}{2}}\sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}

MULTIPLY sample proportion ± inverse confidence level /2 ‘s z-value & square root of sample proportion * (1 - sample proportion) divided by sample size