Chapter 7-9 Study Guide (Stat 101)

Nonparametric

assumptions freeer

Nonparametric tests

The combination of being able to use small sample sizes, makes nonparametric tests desirable for analyzing categorical data obtained from small sample

Degrees of freedom

Has to do with how peaked (or flat) the sampling distribution is for any given statistical test

df = N-1

• Two samples subtract 1 twice

• As SS increases df will increase

Estimation

A process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter.

Point estimates

A sample statistic used to estimate the exact value of a population parameter.

Ex: 13.71 years of edu

Interval Estimates

A sample statistic used to estimate a range of values within which the population parameter may fall.

Aka Confidence interval

Ex: 12-14 years of education

Confidence Interval

A range of values (the raw score) is defined by the confidence level within which the population parameter is estimated to fall.

Defined in terms of confidence level (90%, 95%, 99%)

Can also be defined in terms of margin of error

Confidence Level

The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter.

Used to evaluate the accuracy of interval estimate

Margin of Error

The radius of a confidence interval

Determining the C.I for means

1. Calculate the S.E

2. Decide on C.L and find the corresponding Z score

3. Calculate the CI

4. Interpret the results (“We can say with ___ confidence ____”)

Standard Error of the Mean

σ/√N

Confidence Interval

C.I = Ȳ ± Z (𝜎y)

Confidence Levs + Corresponding Z score

90%: ± 1.65

95% ± 1.96

99%: ± 2.58

Reducing Risk by Increasing Confidence

Larger confidence level = wider range

Trade off

precision decreases (C.I widens)

SS and C.L

As SS increases

• Width of C.I decreases

• Precision of C.I increases

Statistical Hypothesis Testing + Assumptions

A procedure used to evaluate hypotheses about population parameters based on sample statistics.

1. All statistical tests assume RANDOM SAMPLING

2. Tests about MEANS assume an INTERVAL-RATIO level of measurement.

3. Either assume that the population is NORMALLY DISTRIBUTED, or that

   the SAMPLE SIZE is larger than 50 (N>50).

Alt Hypthoesis

Claim to be tested; challenges status quo

H₁: <, >, ≠,

One tailed test

Alt hypothesis is directional

< or > specified value

Right Tailed Test

Pop mean is > specified value

H₁: M >

Left Tailed Test

Pop mean is < specified value

H₁: M <

Two Tailed Test

Pop mean isn’t equal to specified value

H₁: ≠

Null Hypothesis

Indicates no significant diff between population and sample means

expressed in population parameters

H0: =, >, <

The Z statistic (obtained)

• Convert the sample mean into a Z score

• Compute test stat

Z = Ȳ-Mȳ/ σ/ √N

P value

Prob that “measured diff” would occur from random chance if null is true

Measures the unsuality/rarity obtained stat is compared to null

• Smaller p-value = more evidence to reject the null

• Larger p value = null is true

Alpha (α)

Level at which the null is rejected

• Usually set at.05, .01, or .001 level

5 Steps in Hypothesis Testing

• Making assumptions and meeting test requirements

• Formulating the null and research hypotheses and stating the alpha

• Selecting the sampling distribution and specifying the test statistic

• Computing the test statistic

• Making a decision and interpreting the results of the test

Type I error

If the null is true and it’s rejected

Type II error

If the null is false and it’s failed to be rejected

T Statistic (Obtained)

• T reps the standard error units that the sample mean is from the hypothesized value of M (assuming null is true)

t = Ȳ-M/ s/ √N

T distribution

Family of curves determined by degrees of freedom

• used when SS is less than 30

Sampling distribution of the difference between means

Variances are =: SȲ1-Ȳ2 = √(N1-1)s1² + (N2-1) s2²/ (N1+N2)-2 √ N1+N2/ (N1)(N2)

t = Ȳ1- Ȳ2/ SȲ1-Ȳ2

Variances are unequal: SȲ1-Ȳ2= √s1²/N1 + s2² / N2

df: (√s1²/N1 + s2² / N2) ² / (√s1²/N1)/ N1-1) + (s2² / N2)/ (N2-1)

t= Ȳ1- Ȳ2/ SȲ1-Ȳ2

Bivariate Analysis

A method designed to detect and describe the relationship between two nominal/ ordinal variables

Cross Tab

A technique for analyzing the relationship between two variables (IV and DV) that’s been organized in a table

Constructing a Bivariate Table

• Lays the distribution of one variable across categories of another variable

• Classify case based on joint scores or two variables

• Think of it as frequency distributions joined together in one table

Computing percentages in Bivariate Table

• Calculate % with each category of the independent variable

• Interpret by comparing % for diff categories of independent variable

Direct casual relationship

When the relationship between two variables cannot be accounted for by other theoretically relevant variables

Spurious relationship

Relationship between two variables in which both IV and DV are influenced by a causally prior variable and there’s no link between them

3 steps for finding relationship

1. Divide the observation into subgroups based on the control variable

2. Reexamine the relationship between the original two variables separately for the control subgroup

3. Compare partial relationships with original bivariate relationship for total group

Conditional Relationships

When a bivariate relationship differs for different control variables

• condition met → relationship holds

• conditions not met → relationship dissapears