Stats Final Cumulative notes

Vocabulary flashcards for exam review.

Exam Structure

Exam 4 consists of 18 questions: 3 True/False, 12 Multiple Choice, and 1 Short Answer (3 parts).

Material Covered

Chapters 10, 11, and a small part of Chapter 12.

Chapter Emphasis

Main focus is the presentation on Chapters 10 and 11.

Confidence Intervals (One Group)

Previous exam material. Different intervals based on whether $c3$ is known or unknown.

Confidence Intervals (Two Groups)

New material includes confidence intervals for two groups (both $c31$ and $c32$).

Population Standard Deviations Known Confidence Interval Formula

ar{x}1 - ar{x}2 ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } + - z{ rac{b1}{2}} imes egin{pmatrix} rac{c31^2}{n1} + rac{c32^2}{n2} \

ar{x}1 and ar{x}2

Mean of Group 1 and Mean of Group 2, respectively.

z{ rac{b1}{2}}

Critical value (z) for desired confidence level.

Zero within range

To determine if zero falls within the calculated range. If zero is excluded, a significant difference exists.

Population Standard Deviations Unknown but Equal Confidence Interval Formula

ar{x}1 - ar{x}2 ext{ } ext{ } + - t{ rac{b1}{2}} imes sp imes egin{pmatrix} rac{1}{n1} + rac{1}{n2} \

Pooled Standard Deviation ($sp$)

sp = rac{(n1-1)s1^2 + (n2-1)s2^2}{n1+n2-2}

Population Standard Deviations Unknown and Unequal confidence interval

This confidence interval addresses cases when variances are not equal.

s_t

Calculated standard deviation.

No Take-Home Component

All questions will be completed during class.

Critical Values Provided

Critical values for normal distributions will be provided.

Independent Samples

Treatment vs. Placebo Groups

Dependent Samples

Pretest/Posttest design using the same population.

Short Answer Expectation

Expect up to 3 parts: hypothesis test for equality of $c31$ and $c32$, followed by the appropriate confidence interval based on the test.

Formulas and Conditions

Master which formulas apply under which conditions.

Sample Data Practice

Practice identifying sample data from tables.

Independent vs. Dependent Samples

Familiarize yourself with concepts through examples.

Hypothesis Tests for Variances

Importance of understanding these tests in statistics.

Confidence Intervals

Used to estimate where population parameters lie based on sample statistics.

Choice of Interval

Dependent on whether standard deviations are known or estimated.

Null Hypothesis ($H_0$)

Assumes no difference in variances.

Alternative Hypothesis ($H_a$)

Assumes there is a difference in variances.

Test Statistic for Variances

L = rac{S{max}^2}{S{min}^2}

$S{max}^2$

Maximum variance.

$S{min}^2$

Minimum variance.

Decision Rule (Variances)

If $L >$ critical value, reject $H0$ (variances are not equal). If $L leq$ critical value, fail to reject $H0$ (variances are equal).

Emphasis on

Practical application of statistical tests in real-world contexts.

Sample

Hypothesis testing and confidence intervals primarily for independent samples.

Exception

Testing for equal variances.

Focus on Confidence Intervals

Paired samples (dependent samples) which often use a pretest-posttest design.

Importance Ensuring Each Observation

Posttest has a corresponding observation in the pretest.

Average of Differences

ar{D} = rac{ ext{sum from }i=1 ext{ to }n D_i}{n}

Sample Standard Deviation for Differences

SD = rac{ ext{sqrt}igg( ext{sum from }i=1 ext{ to }n (Di - ar{D})^2igg)}{n - 1}

Confidence Interval Calculation for Dependent Samples

ar{D} ext{ ± } t{ rac{ ext{α}}{2}} rac{SD}{ ext{sqrt}(n)}

Objective

Calculation of test statistics based on sample differences and comparison with critical values focusing on a two-tailed test framework.

Formula Conditions

n1p1, n1(1-p1), n2p2, n2(1-p2) must be greater than or equal to 5 for validity.

Proportion Confidence Interval Formula

ar{p1} - ar{p2} ext{ ± } Z{ rac{ ext{α}}{2}} ext{sqrt}igg( rac{ar{p1}(1-ar{p1})}{n1} + rac{ar{p2}(1-ar{p2})}{n_2}igg)

ANOVA

Analysis of Variance allows comparison of means across three or more groups instead of two.

ANOVA Null Hypothesis

All means are equal across multiple groups.

ANOVA Components

SSB, SSW, SST, degrees of freedom, MSB, MSW, and F-statistic, each component plays a key role in hypothesis testing.

Assignment

This assignment covers the material from Chapters 10, 11, and 12.

Formula: Calculate 95% Confidence Interval

(\bar{x}1 - \bar{x}2) \pm z{\alpha/2} \cdot \sqrt{\frac{\sigma1^2}{n1} + \frac{\sigma2^2}{n_2}}

Pooled Standard Deviation Formula

sp = \sqrt{\frac{(n1 - 1)s1^2 + (n2 - 1)s2^2}{n1 + n2 - 2}}

Part D: Conduct Hypothesis Test Null Hypothesis

H0: \sigma1^2 = \sigma_2^2

Alternative

Not all means are equal.

Test Statistic

Test statistics are computed using sums of squares, mean squares.

If p < α

Reject null hypothesis.

Hypothesis Statement

Remember the structure and flow of each statistical test: hypothesis statement, test statistic computation, and result interpretation.

Step-by-Step Process for Short Answer Questions: Null Hypothesis

e.g., ext{sigma}1^2 = ext{sigma}_2^2

Step-by-Step Process for Short Answer Questions: Alternative Hypothesis

e.g., ext{sigma}1^2 eq ext{sigma}_2^2

Step-by-Step Process for Short Answer Questions: Calculate Test Statistic Formula

rac{s{ ext{max}}^2}{s{ ext{min}}^2}

If variances are equal

Use the pooled variance formula.

If variances are not equal

Use the separate variance formula.

For ANOVA

Analyze whether all means are equal across multiple groups.

Paired Samples

Same subjects measured before and after an intervention.

Conditions for Normal Distribution in Proportions

n1 p1 ext{ and } n2 p2 ext{ both } ext{≥} 5 and n1 (1 - p1) ext{ and } n2 (1 - p2) ext{ both } ext{≥} 5

P-value

P-value signals.

F-stat

Calculated only when testing if variances are equal.

An easy way to differentiate

if the critical value is provided, it is a Z-stat.

The final exam covers

material up to a certain point.

Exam is the

shortest one of the semester, comprising 18 questions.

Main topics

include confidence intervals and hypothesis tests from chapters 10 and 11.

No take-home exam

included.

Independent Samples

Two different groups (e.g., treatment vs. control).

Increased Confidence Level

Results in an increased margin of error.

Decreased Confidence Level

Results in a decreased margin of error.

Confidence interval and test stat formulas

For when standard deviations are known are on the formula sheet.

Confidence interval and test stat formulas are provided

When standard deviations are unknown and unequal.

Confidence interval and test stat formulas

When standard deviations are unknown and equal.

Alternative Hypothesis

A statement that contradicts the null hypothesis. Researchers aim to find evidence supporting the alternative hypothesis.

Critical Value

A predefined threshold used in hypothesis testing to decide whether to reject the null hypothesis. If the test statistic exceeds the critical value, the null hypothesis is rejected.

Significance Level (alpha)

The probability of rejecting the null hypothesis when it is true. It is often set at 0.05, corresponding to a 5% risk of making a wrong decision.

Conclusion

The final decision made in hypothesis testing, based on the comparison of the test statistic and critical value. It may involve rejecting or failing to reject the null hypothesis.