Confidence Intervals and Variable Analysis Study Notes

SECTION 3.3: Confidence Intervals for Single Means

1. Purpose of Confidence Intervals

  • To find a likely range of values for a parameter.

2. General Forms

  • General formula for a confidence interval:
    ext{Parameter} ext{ (Point Estimate)} ext{ ± } ext{Margin of Error}

    • Margin of Error formula:
      ext{Margin of Error} = M imes S

  • 95% Confidence Interval for Proportions:
    p ext{ ± } 2 imes rac{p(1-p)}{n}

  • Question: What do you think changes in a 95% Confidence Interval for Means?

3. Key Elements of Confidence Intervals

  • General formula for means: ar{X} ± M_m imes S
    • Where:
    • ar{X} = sample mean
    • S = standard deviation
  • Conditions for validity:
    • Sample size must be greater than 20.
    • Distribution should not be strongly skewed.

3.1 Standard Error

  • Standard error represents the standard deviation of the sampling distribution.
  • Note: It is the same as what was used in standardized statistics in Chapter 2.
    SE = rac{S}{ ext{sqrt}(n)}

4. 95% Confidence Interval

  • Multiplier of 2 is used for 95% confidence intervals: C_{95 ext{%}} = ar{X} ± 2 imes SE
    • This results in the interval:
      (LL, UL)
    • Where:
    • L_L = lower limit
    • U_L = upper limit

5. Example Analysis

  • Given data from a sample:
    • Two Cal Poly students gathered data on prices for a random sample of 30 textbooks:
    • Average price: ar{X} = 65.02
    • Standard deviation: S = 51.42
    • Interpretation of confidence interval:
    • "We are confident that the long run __ price of textbooks at Cal Poly falls between ____ and ____”.

6. Interpretation of Confidence Interval

  • When analyzing confidence intervals:
    • If the null hypothesis value is inside of the interval, it is considered a likely value, leading to a decision to Fail to Reject.
    • If the null hypothesis value is outside of the interval, it is deemed NOT a likely value, leading to a decision to Reject.
  • Conclusion in context:
    • We either do or do not have evidence to conclude a specific alternative hypothesis.

SECTION 3.4: Width of Confidence Interval

1. Factors Affecting Confidence Interval Width

  • Key factors:
    • Confidence Level
    • Standard Error
    • Sample Size

2. Relationship Between Confidence Level and Width

  • Significance Level + Confidence Level = 100%
  • If confidence level increases, the significance level decreases:
    • Results in rejecting less often and widening the interval.
  • If confidence level decreases, the significance level increases:
    • Results in rejecting more often and narrowing the interval.
  • Conclusion: "If you want to be more confident you have to cover more values."

3. Relationship Between Sample Size and Width

  • Sample Size Increases:
    • Less variability, leading to a narrower interval.
  • Sample Size Decreases:
    • More variability, leading to a wider interval.
  • Conclusion: "Larger sample size makes you more confident in your answer."

4. General Formulas for Width

  • General confidence interval formula:
    ext{Confidence Interval} = ext{Estimation} ext{ ± } M_m imes SE
  • Effects of variability:
    • If standard error increases, margin of error increases consequently widening the interval.
    • If standard error decreases, margin of error decreases consequently narrowing the interval.

SECTION 4: Explanatory, Response, and Confounding Variables

1. Warm-Up Questions

  • Think of two things that are “associated”:
    • Discussion prompt on association.

2. Learning Objectives

  • Distinguish between explanatory, response, and confounding variables.
  • Calculate conditional probabilities and explain their meaning.
  • Introduction to the concepts of cause-and-effect and association.

3. Definitions

  1. Explanatory Variable:
    • The variable thought to be “explaining” the change.
  2. Response Variable:
    • The variable which is being changed or impacted.
  3. Confounding Variable:
    • A variable related to both explanatory and response variables, where its effects on the response variable cannot be separated.
  4. Cause and Effect:
    • Establishing that one event is the direct result of another.
  5. Association:
    • One variable provides information about another.

4. Examples of Explanatory and Response Variables

  • Situation A:

    • Vehicles: Toyota Camry (34 MPG), Ford Fusion (26 MPG), Honda Accord (36 MPG), Dodge Dart (21 MPG)
    • Explanatory variable: Make/Model of car
    • Response variable: Miles per gallon.

  • Situation B:

    • Apartment complexes: Good View Apartments (7.8/10), Nice High Rises (8.9/10), Down Dumps Slums (3.2/10)

  • Situation C:

    • People with hydration ratings: Kendra (90/100), Riddy (61/100), Thomas (76/100).

5. Potential Confounding Variables

  • Recall Situation A: Change in vehicle model year might serve as a confounding variable.

6. Conditional Probability Questions

  • Analyze specific situations with given data:
    • Examples include analyzing win-loss records of coaches and ranking results based on performance.

7. Exit Ticket Scenario

  • Researcher finds a relationship between ice cream sales and shark attacks; identify the explanatory variable, response variable, and a potential confounding variable.