standard error

Statistical Inference

using data from a sample to draw conclusions about a population

Parameter

number that describes a population

Statistic

a number calculated from the sample

Sample Proportion

the statistic that estimates the parameter p

Sample Proportion Equation

p'(hat) = count of successes in sample / n

If the sample size is large enough, then . . .

1. Sample Distribution = ~Normal

2. Mean = p(hat) of p

3. Standard Deviation = √p(1-p)/n

95% Confidence Interval

an interval calculated from sample data by a process that is guaranteed to capture the true population parameter in 95% of all samples

Sampling Distribution

the distribution of values by the statistic in all possible samples

Sampling Distribution

a distribution of a statistic that tells us what values the statistic takes in repeated samples from the same population and how often it takes those values

Standard Error

the standard deviation of the sampling distribution of a sample statistic

Standard Errors Interval

p(hat) ± 2√p(1-p)/n

95% Confidence Interval for a Population Proportion

p(hat) ± 2√p(hat)(1-p(hat))/n

Level C Confidence Interval has two parts:

1. Interval calculated from the data

2. Confidence Level C, which gives the probability that the interval will capture the true parameter value in repeated samples

Critical Values z*

in any Normal Distribution, there is area (probability) C under the curve between -z* and +z* standard deviations away from the mean

C Confidence Interval for p

p(hat) ± z*√p(hat)(1-p(hat))/n

Sampling Distribution of a Sample Mean

1. Sampling distribution of x(hat) is ~Normal when the sample size n is large

2. Mean of the sampling distribution of x(hat) is equal to µ

3. Standard deviation or Standard error of the sampling distribution of x(hat) is σ/√n

Properties of the Sample Mean x(hat)

1. Mean of a # of observations is less variable than individual observations

2. Distribution of a mean of a # of observations is more Normal than the distribution of individual observations

Central Limit Theorem

theory that as we take more and more observations at random from any population, the distribution of the mean of these observations eventually gets close to a Normal distribution

Level C Confidence Interval for µ

x(hat) ± z* (s/√n)

Test of Significance

• test designed to assess the evidence for some claim about the value of an unknown parameter

• test designed to assess the strength of the evidence against the null hypothesis

null hypothesis (H₀)

the claim being tested in a statistical test that is designed to assess the strength of the evidence against the claim

• statement of “no effect” or “no difference”

Alternative Hypothesis (Hₐ)

statement that researchers suspect, or hope, to be true

P-value

probability (assuming H₀ is true) that the sample outcome would be as extreme or more extreme than the actually observed outcome

• the smaller the P-value, the stronger the evidence against H₀

Statistical Significance at Level ⍺

when the P-value is as small or smaller than ⍺

Significance Level

the decisive value of P, by which we determine that a sample result is statistically significant if it would occur just by chance no more than his percentage of the time in repeated samples

Test Statistic

the standard score computed based on the sample data

Two major types of statistical inference:

1. Confidence Intervals

2. Significance Tests

For a Confidence Interval & Test for a Proportion p

1. Data must be a Simple Random Sample (SRS) from the population of ibterest

2. Dropouts & Nonresponse are important sources of error