Stats Final Cumulative notes

Exam Structure

• Exam Number 4 consists of 18 questions:
  • 3 True/False questions
  • 12 Multiple Choice questions
  • 1 Short Answer question with three parts

Material Covered

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

Chapter Presentations

• Current presentation combines Chapters 10 and 11
• Last presentation for Chapter 12 has minimal content (8 slides)
• Emphasis: The main focus for exam preparation is the presentation on Chapters 10 and 11

Confidence Intervals Recap

• Previous exam included confidence intervals with one group:
  • Different intervals depending on whether population standard deviation (\sigma) was known or unknown.
• New material includes confidence intervals for two groups.
• Key types of confidence intervals for two groups:
  • Both \sigma1 and \sigma2 known:
    • Formula: \bar{x}1 - \bar{x}2 \pm z{\frac{\alpha}{2}} \times \sqrt{\frac{\sigma1^2}{n1} + \frac{\sigma2^2}{n_2}}
      • \bar{x}1 = Mean of Group 1, \bar{x}2 = Mean of Group 2, z_{\frac{\alpha}{2}} = critical value (z) for desired confidence level.
      • To determine if zero falls within the calculated range. If zero is excluded, a significant difference exists.
  • \sigma1 and \sigma2 unknown but equal:
    • Formula: \bar{x}1 - \bar{x}2 \pm t{\frac{\alpha}{2}} \times sp \times \sqrt{\frac{1}{n1} + \frac{1}{n2}}
      • This uses the pooled standard deviation (sp) = sp = \sqrt{\frac{(n1-1)s1^2 + (n2-1)s2^2}{n1+n2-2}}
  • \sigma1 and \sigma2 unknown and unequal:
    • Formula: \bar{x}1 - \bar{x}2 \pm t{\frac{\alpha}{2}} \times st \times \sqrt{\frac{1}{n1} + \frac{1}{n2}}
      • s_t = calculated stand.dev.
      • This confidence interval addresses cases when variances are not equal.

Exam Additional Reminders

• No take-home exam component: All questions will be completed during class
• Critical values for normal distributions will be provided; no calculations needed on this portion.
  • For example:
    • 99%: z = 2.576
    • 95%: z = 1.96
    • 90%: z = 1.645
• Understanding of independent vs. dependent samples will be tested:
  • Examples of independent samples: Treatment vs. Placebo Groups
  • Examples of dependent samples: Pretest/Posttest design using the same population.

Short Answer Questions

• Anticipate up to 3 parts for one short answer question that involves:
  • Hypothesis test to determine if \sigma1 and \sigma2 are equal
  • Followed by conducting the appropriate confidence interval depending on results of hypothesis test.

Key Takeaways

• Master which formulas applies under which conditions before the exam
• Practice identifying sample data from tables
• Familiarize yourself with concepts of independent vs. dependent samples through examples provided in class.

Introduction to Hypothesis Testing for Variances

• Importance of understanding hypothesis tests for variances in statistics.

Key Concepts

• Confidence Intervals: Used to estimate where population parameters lie based on sample statistics.
  • Three types of confidence intervals discussed:
    1. Population standard deviations known.
    2. Population standard deviations unknown and equal.
    3. Population standard deviations unknown and unequal.
  • Choice of interval dependent on whether standard deviations are known or estimated.

Hypothesis Testing Steps

1. State the Null and Alternative Hypotheses:
   • Null Hypothesis (H_0): Assumes no difference in variances.
   • Alternative Hypothesis (H_a): Assumes there is a difference in variances.
2. Calculate the Test Statistic:
   • For variances, the test statistic used is the L statistic, defined as: L = \frac{S{\text{max}}^2}{S{\text{min}}^2}
     • where S{\text{max}}^2 is the maximum variance and S{\text{min}}^2 is the minimum variance.
3. Determine Critical Values:
   • Critical values are provided for the test and allow comparison with the L statistic.
4. Decision Rule:
   • If L > critical value, reject H_0 (variances are not equal).
   • If L \leq critical value, fail to reject H_0 (variances are equal).
5. Draw a Conclusion:
   • Based on the hypothesis test results, conclude if variances are equal or not.

Important Formulas

• Test Statistic Formula:
  • Test statistic for variances: L = \frac{S{\text{max}}^2}{S{\text{min}}^2}
• Critical Values: Provided for specific significance levels (1%, 5%, 10%).

Examples

• Examples for calculating test statistics, determining critical values, and interpreting results based on sample data.
• Discussed real-world implications of hypothesis test outcomes, such as differences in damage from car bumpers based on group samples.
• Example with Power Five vs Non Power Five schools for NIL valuations, demonstrating statistical significance with calculated test statistics.

Conclusion

• Understanding these tests aids in assessing differences in means or variances in various scenarios.
• Emphasis on practical application of statistical tests in real-world contexts.
• Reminder: All exam related calculations and tests will be provided to students during assessments.

Class Schedule

• Last class meets on Tuesday, not Thursday due to the schedule.
• No question day planned in the last week due to class schedule change.
• Agenda for the next classes:
  • Finish current presentation.
  • Present Chapter 12.
  • Assignment on Thursday.
  • Review before final class.

Overview of Previous Material

• Previous discussions centered on hypothesis testing and confidence intervals primarily for independent samples.
• One exception addressed was testing for equal variances.

Introduction to Dependent Samples

• Focus on confidence intervals for paired samples (dependent samples) which often use a pretest-posttest design.
• Example: Comparing a pretest score to a posttest score for the same group of students.
• Importance of ensuring each observation in the posttest has a corresponding observation in the pretest.

Calculating Differences in Paired Samples

• Define disparity between pretest and posttest scores as Di = Xi - Y_i, where
  • X_i = posttest score
  • Y_i = pretest score
  • Average of differences: \bar{D} = \frac{\sum{i=1}^{n} Di}{n}

Sample Standard Deviation for Differences

• Formula: SD = \sqrt{\frac{\sum{i=1}^{n} (Di - \bar{D})^2}{n - 1}}
  • Example calculation for each individual observation’s difference from the mean difference.

Confidence Interval Calculation for Dependent Samples

• Formula: \bar{D} \pm t_{\frac{\alpha}{2}} \frac{SD}{\sqrt{n}}
  • Key elements needed: Average differences, critical t-value, and sample standard deviation.

Example Problem: Gas Mileage and Fuel Type

• The scenario: An oil company investigates the miles per gallon between nonethanol and ethanol gas by using the same drivers for tests.
• Calculating the differences for each observation, followed by compiling into a table for analysis.
• Successful calculation of sums and average differences to compute confidence intervals and hypothesis tests.

Hypothesis Testing for Means

• Formulating null and alternative hypotheses concerning fuel efficiencies.
• Calculation of test statistics based on sample differences and comparison with critical values focusing on a two-tailed test framework.
• Referential use of provided critical values to draw conclusions (reject or fail to reject null hypothesis).

Transition to Differences Between Proportions

• Significance in proportions when measuring comparisons between two group means with defined conditions (e.g., success rate must be calculated).
• Important formula conditions:
  • n1p1, n1(1-p1), n2p2, n2(1-p2) must be greater than or equal to 5 for validity.
• Proportion confidence interval formula: \bar{p}1 - \bar{p}2 \pm Z{\frac{\alpha}{2}} \sqrt{\frac{\bar{p}1(1-\bar{p}1)}{n1} + \frac{\bar{p}2(1-\bar{p}2)}{n_2}}
• Example related to reliability of hairdryer units showcasing percentage failure rates with critical implications for production decisions.

Introduction to ANOVA (Analysis of Variance)

• Significance: ANOVA allows comparison of means across three or more groups instead of two.
• Null hypothesis example for ANOVA includes that all means are equal across multiple groups.
• Different calculations needed compared to prior tests focusing on group variance analysis (between vs. within groups).

ANOVA Table and Value Calculations

• Each component (SSB, SSW, SST, degrees of freedom, MSB, MSW, and F-statistic) plays a key role in hypothesis testing:
  • Total observations impact overall variability calculations as does the number of groups.
  • Connection of group findings to overall research conclusions shown through statistical tables and output from software tools.
  • In-class exercise involves filling out an ANOVA table based on given data and ensuring each component's calculations corroborate logically (i.e., SSB + SSW = SST).

Final Review and Preparations

• Classes to include practical application through previously noted examples to solidify understanding.
• Allocation of time to review practice problems to better prepare before final examinations, focusing on critical values and hypothesis testing mechanisms.

Assignment Overview

• This assignment covers the material from Chapters 10, 11, and 12.
• One overarching assignment for exam preparation.

Confidence Interval for Two Sample Means Problem 1: Data and Initial Setup

• Sample Statistics:
  • \bar{x}_1 = 15
  • \bar{x}_2 = 40
• Population Standard Deviations:
  • \sigma_1 = 5
  • \sigma_2 = 4
• Sample Sizes:
  • n_1 = 40
  • n_2 = 40

Part A: Calculate 95% Confidence Interval

• Formula: (\bar{x}1 - \bar{x}2) \pm z{\alpha/2} \cdot \sqrt{\frac{\sigma1^2}{n1} + \frac{\sigma2^2}{n_2}}
• Critical Value for 95% CI: 1.96
• Calculation Steps:
  • Find the difference in means: \bar{x}1 - \bar{x}2 = 15 - 40 = -25
  • Calculate the standard error:
    • \sqrt{\frac{5^2}{40} + \frac{4^2}{40}} = \sqrt{\frac{25}{40} + \frac{16}{40}} = \sqrt{1.025} \approx 1.012
  • Apply the formula:
    • -25 \pm 1.96 \cdot 1.012
    • Lower limit: -25 - 1.984 = -26.984
    • Upper limit: -25 + 1.984 = -23.016
• Confidence Interval: [-26.984, -23.016]

Part B: Population Standard Deviation Unknown

• Sample Standard Deviations:
  • s_1 = 5
  • s_2 = 4
• Sample sizes updated: n1 = 10, n2 = 10
• Use pooled standard deviation, sp, as follows: sp = \sqrt{\frac{(n1 - 1)s1^2 + (n2 - 1)s2^2}{n1 + n2 - 2}}
• Calculate:
  • s_p = \sqrt{\frac{(10-1)(5^2) + (10-1)(4^2)}{10 + 10 - 2}} = \sqrt{17.187} \approx 4.129
• Calculate CI:
  • Formula: \bar{x}1 - \bar{x}2 \pm t{\alpha/2} \cdot sp \cdot \sqrt{\frac{1}{n1} + \frac{1}{n2}}
  • Use t_{\alpha/2} = 2.1009 and calculate:
    • 10 \pm 2.1009 \cdot 1.303 \rightarrow \text{results in interval}[5.810, 14.190]

Part C: Variances Not Equal

• Derive test stat and use new critical value t_{\alpha/2} = 2.1098:
• Follow similar steps and calculate:
  • Components of CI similar to previous parts.

Part D: Conduct Hypothesis Test

• Null Hypothesis: H0: \sigma1^2 = \sigma_2^2
• Test statistic: F = \frac{s{\text{max}}^2}{s{\text{min}}^2}
  • Find maximum and minimum variances from previous calculations.
  • Compare: Test needed to determine if null hypothesis is rejected.
• Critical value: F{\text{critical}} = 4.026; determine if F < F{\text{critical}}

Hypothesis Test for Difference of Means

• Perform 5% significance level tests for hypotheses.
• Calculate test statistics based on difference in sample means.

Problem 2: ANOVA Tests

• Review tools used by programmers and time taken for different tools.
• Null Hypothesis: All means are equal across tools.
• Alternatives: Not all means are equal.
• Test statistics are computed using sums of squares, mean squares.

Using p-values

• Decision Rule considering p-values:
  • If p < \alpha, reject null hypothesis.
• Attention: Interpret results based on values of test statistics and corresponding p-values.

Additional Notes

• Remember the structure and flow of each statistical test: hypothesis statement, test statistic computation, and result interpretation.
• Maintain familiarity with formula sheets and critical values per significance level, as they will be provided in the test.

Overview of the Exam Structure

• Total Questions: 18
  • 3 Multiple Choice Questions
  • 3 Short Answer Questions
• No Take-Home Issues: All questions will be on the exam itself.

Exam Date and Preparation

• Exam Date: Monday, May [Specific Date], at 12:00 PM.
• Final Class Meeting: April 29 (next Tuesday); will be a Q&A session with no new content.
  • It's advised to review material over the weekend and prepare questions for the final class.

Study Guide Overview

• Focus Areas: Problems from Assignment Number 10 are pivotal for the exam.
  • All highlighted problems are relevant.
  • Specific parts of problems dealing with calculating degrees of freedom and critical values can be ignored for this exam.

Critical Values to Remember

• You only need to memorize three two-tailed critical values:
  • 1% Significance Level: z = 2.576
  • 5% Significance Level: z = 1.96
  • 10% Significance Level: z = 1.645
• Important Note:
  • Although critical values might show as 2.575 in tables, stick to 2.576 in calculations.

Hypothesis Testing

• Step-by-Step Process for Short Answer Questions:
  • Null Hypothesis (H0): e.g., \sigma1^2 = \sigma_2^2
  • Alternative Hypothesis (Ha): e.g., \sigma1^2 \neq \sigma_2^2
  • Calculate Test Statistic: Formula: \frac{s{\text{max}}^2}{s{\text{min}}^2}
  • Utilize the larger sample variance in the numerator.
  • Compare Test Statistic to Critical Value provided in the exam.
  • Conclusion:
    • If test statistic > critical value, reject H_0.
    • If test statistic ≤ critical value, fail to reject H_0.
• Note: Always show work for each of the five steps to receive full credit.

Confidence Intervals Based on Variances

• Use different formulas based on the equality of variances:
  • If variances are equal, use the pooled variance formula.
  • If variances are not equal, use the separate variance formula.

ANOVA and Multiple Choice Questions

• You will encounter multiple-choice questions that require hypothesis testing or confidence interval calculations using provided data.
  • For ANOVA, you will analyze whether all means are equal across multiple groups.

Differences Between Sample Types

• Independent Samples: Two different groups (e.g., treatment vs. control).
• Paired Samples: Same subjects measured before and after an intervention.

Key Concepts in Confidence Intervals

• Recognize the inverse relationship between confidence level and margin of error:
  • Increasing confidence level increases margin of error.
  • Decreasing confidence level decreases margin of error.

Conditions for Normal Distribution in Proportions

• Ensure these criteria are satisfied for a normal distribution when doing hypothesis tests for proportions:
  • n1 p1 \text{ and } n2 p2 \text{ both } \geq 5
  • n1 (1 - p1) \text{ and } n2 (1 - p2) \text{ both } \geq 5

Summary of Important Notes for Exam

• Memorize critical values for two-tailed tests.
• Be prepared for hypothesis testing: both equal variances and calculating confidence intervals.
• Understand the types of samples for the context of your testing.
• Review your notes and previous assignments thoroughly before the exam.

P-Value and F-Stat

• P-value signals.
• P-value can be calculated, but it is not required.
• Example: If a value is less than or equal to 0.01, it should be indicated as such on the test, such as 2.575.
• F-stat is calculated only when testing if variances are equal.
• T-stat and Z-stat are used for other scenarios.

T-Stat vs. Z-Stat

• The choice between T-stat and Z-stat depends on whether population standard deviations are known and equal, unknown and equal, or unknown and unequal.
• If standard deviations are unknown and equal, or unknown and unequal, T-stat should be used.
• An easy way to differentiate: if the critical value is provided, it is a Z-stat. Critical values for Z-stats must be memorized and usually not provided.

Formula Sheet

• Confidence interval and test stat formulas for when standard deviations are known are on the formula sheet.
• Formulas for when standard deviations are unknown and equal are also provided, indicated by "ST".
• Confidence interval and test stat formulas are provided when standard deviations are unknown and unequal.

Exam Details and Schedule

• The final exam covers material up to a certain point.
• The exam can potentially be taken on Tuesday if needed.
• Location for taking the exam is in rooms 305 to 307.
• If arriving late to the test (e.g., 3:30 PM), be aware that the test concludes at 4:00 PM.

Assignment Submission

• Assignments can be turned in the next day.

Exam Format

• The exam is the shortest one of the semester, comprising 18 questions.
• There are three true/false questions, 12 multiple-choice questions, and three short answer questions.
• Each question weighs a little more due to the exam's length.

Exam Content

• Main topics include confidence intervals and hypothesis tests from chapters 10 and 11.
• Topics include:
  • Confidence interval and hypothesis test for when standard deviations are known.
  • Confidence interval and hypothesis test when standard deviations are unknown and equal.
  • Confidence interval and hypothesis test when standard deviations are unknown and unequal.
  • Confidence interval and hypothesis test for the proportion.
  • Confidence interval and hypothesis test for paired samples (all from chapter 10).
  • Test for whether variances are equal (from chapter 11).
  • Chapter 12 content.
• There is significantly less information covered on this exam compared to previous ones.

Grading Policy

• No take-home exam.
• It is possible to achieve an A in the course even with a zero on the final exam, depending on current grades.

Grade Calculation Example

• Midterm score and the required final exam score to achieve a specific grade.
• Rounding policy used for grades (e.g., .49 rule).

Course Evaluations

• Course evaluations are open and will close soon.

Personal Anecdotes

• Experiences with taking multiple exams in a short period.
• Story about receiving and taking the wrong test for 30 minutes due to similarity between tests for different classes.