STAT 124 - Analysis of a Single Sample

Introduction to Statistics

Course Information

• Course Code: STAT 124

• Institution: University of British Columbia (UBC)

• Focus: Analysis of a Single Sample

Analysis of One Sample - Example

Problem Statement

• Sample Size: 200 university students.

• Observations: 31 students are business majors, 169 others are not.

Data Summary

Observed Data

• Total Sample Size ($n$): 200

• Business Majors: 31

• Other Majors: 169

• Sample Proportion ($\, \, ar{p}$): $\bar{p} = rac{31}{200} = 0.155$ or 15.5%

Standard Error and Margin of Error

Calculation of Standard Error (SE)

• Standard Error Formula: $SE = ext{sqrt} rac{\bar{p}(1 - \bar{p})}{n}$

• Calculation:

  • $SE = ext{sqrt} rac{0.155 imes 0.845}{200} ext{≈} 0.0256$

Margin of Error (ME) for 95% Confidence Level

• Critical Value ($z$): 1.96

• Margin of Error Formula: $ME = z imes SE$

• Calculation:

  • $ME = 1.96 imes 0.0256 ext{≈} 0.0501$

Confidence Interval Interpretation

Calculation of 95% Confidence Interval

• Confidence Interval Formula: $CI = \bar{p} ext{±} ME$

• Calculation:

  • $CI = 0.155 ext{±} 0.050 ext{→} [0.105, 0.205]$

Interpretation

• We are 95% confident that the true proportion of business majors in the entire university population lies between 10.5% and 20.5%.

Defining the Standard Error

Conceptual Definition

• Standard Error (SE) measures the dispersion of sample proportions around the true population proportion.

• Significance:

  • Quantifies sampling error.

  • Indicates how much we expect our sample proportion ($ar{p}$) to miss the true population proportion ($p$).

  • Represents the standard deviation of the sampling distribution.

Mathematical Formula for Standard Error

Formula Overview

• Standard Error Formula: $SE = ext{sqrt} rac{p(1-p)}{n}$

• Components:

  • $p$: Proportion of successes.

  • $1 - p$: Proportion of failures.

  • $n$: Sample size (number of observations).

The Law of Large Numbers

Importance of Sample Size

• Larger sample sizes lead to decreased Standard Error because $n$ is in the denominator.

• Example: To halve the Standard Error, you must quadruple the sample size.

Precision vs. Accuracy

• A small Standard Error indicates high precision, suggesting that multiple samples will yield similar results.

The Concept of Interval Estimation

Definitions

• Point Estimate: A single value (e.g., 51%) used to estimate a population parameter.

• Problem with Point Estimates: Samples vary, making point estimates likely to be slightly inaccurate.

• Solution: Use an Interval Estimate to provide a range of plausible values.

• Definition of Confidence Interval: A range derived from sample data likely to contain the unknown population parameter.

Constructing the Confidence Interval

Construction Methodology

• The Confidence Interval ($CI$) is constructed by adding and subtracting a Margin of Error from the sample proportion ($ar{p}$): $CI = \bar{p} ext{±} z^* SE$

Trade-offs in Construction

• Confidence Level: Increasing the confidence level (e.g., from 95% to 99%) results in a wider interval.

• Sample Size: Increasing $n$ narrows the interval (greater precision).

Correct Interpretation of Confidence Intervals

Understanding 95% Confidence

• Interpretation: “95% of all possible samples of this size will produce an interval that captures the true population proportion.”

Common Misconceptions

• Incorrect Interpretation: “There is a 95% probability the true proportion is between A and B.” (The true proportion is fixed, not a variable)

• Correct Interpretation: “We used a process that is successful 95% of the time.”

Hypothesis Testing Approach

Application of Hypothesis Testing

• This data can be utilized to test claims about the population.

• Null Hypothesis ($H0$): $p = p0$ (e.g.,

  • The university claims 20% of students are business majors).

• Alternative Hypothesis ($Ha$): $p eq p0$.

The Research Question

Claim vs. Observed Data

• University Claim: 20% business majors ($p_0 = 0.20$).

• Observed: In our sample of $n = 200$, 15.5% ($ar{p} = 0.155$).

Hypotheses Formulation

• Null Hypothesis ($H_0$): $p = 0.20$

• Alternative Hypothesis ($H_a$): $p
  eq 0.20$

Calculating the Z-Statistic

Formula and Calculation Steps

• Z-Statistic Formula: $z = rac{\bar{p} - p<em>0}{ ext{SE}</em>{0}}$

• Steps:

  1. Difference calculation: $0.155 - 0.20 = -0.045$

  2. Standard Error calculation:

  • $ext{SE}_{0} = ext{sqrt} rac{0.20 imes 0.80}{200} ≈ 0.0283$

  1. Compute $z$: $z = rac{-0.045}{0.0283} = -1.59$

Statistical Decision

Results Overview

• Calculated $z$: -1.59

• Critical $z$ (at $ ext{α} = 0.05$): ±1.96

Conclusion

• Decision: Fail to Reject $H_0$

• Interpretation: Since $|-1.59| < 1.96$, the result is not statistically significant. There is insufficient evidence to claim the actual proportion differs from 20%.

Practical Significance vs. Statistical Significance

Insights

• Despite the result not being statistically significant, a 4.5% gap might be important for university planning.

• Increasing Power: If the same 15.5% result comes from a sample of $n = 500$, the resulting $z$ would be -2.51, which is significant ($p < 0.05$).

A More Complicated Situation - Analysis of Preferences

Example Scenario

• Random sample of 100 people asked about their preference between Starbucks and Tim Horton’s.

• Responses:

  • Starbucks: 51

  • Tim Horton’s: 39

  • No Preference: 10

Inquiry

• Question posed: How to analyze these data?

Study Overview and Data Distribution

Sample Details

• Random Sample Size ($n$): 100

• Population: Large general population

Summary of Survey Responses

• Starbucks: 51 (51%)

• Tim Horton’s: 39 (39%)

• No Preference: 10 (10%)

• Total: 100 (100%)

Methodological Approaches for Analysis

Approaches to Analyze Categorical Frequency Data

1. Chi-Square Goodness-of-Fit Test:

   • Tests if distribution across categories differs from theoretical expectation (e.g., an even distribution).

2. Z-Test for Proportions:

   • Used for direct comparison between two specific groups (e.g., Starbucks vs. Tim Horton’s), with the third group as an external variable.

Precision: The Confidence Interval Analysis

Calculation of Confidence Interval for Starbucks Preference

1. Standard Error (SE):

   • $SE = ext{sqrt} rac{p(1 - p)}{n} = ext{sqrt} rac{0.51 imes 0.49}{100} ext{≈} 0.05$

2. Margin of Error (ME):

   • At 95% confidence ($z = 1.96$):
     $ME = 1.96 imes 0.05 ext{≈} 0.10$ (or 10%)

3. Confidence Interval Interpretation:

   • We are 95% confident that the true population preference for Starbucks lies between (51% - 10%) 41% and (51% + 10%) 61%.

Key Insights and Interpretation

Analysis Insights

• The Lead: Starbucks holds a 12-point lead over Tim Horton’s in this sample.

• Neutral Variable Impact: The 10% ”No Preference” group indicates high brand engagement in this population.

• Statistical Power Consideration: With $n = 100$, the margin of error (±10%) is considerable. A critical note is that lower bound of Starbucks CI (41%) overlaps with Tim Horton’s result (39%), necessitating inferential testing (like Chi-Square) to affirm significance.

More Detail on the Chi-Square Approach

Data Overview for Chi-Square Test

• Total Sample Size ($n$): 100

• Preference Observations:

  • Starbucks: 51%

  • Tim Horton’s: 39%

  • No Preference: 10%

The Research Question

• Is the observed preference for Starbucks statistically significant compared to a neutral distribution?

The Chi-Square Goodness-of-Fit Test

Null Hypothesis Formulation

• Null Hypothesis ($H_0$): All preferences are equal.

• Chi-Square Test Formula: $ext{χ}^2 = ext{Σ} rac{(O<em>i - E</em>i)^2}{E_i}$

  • Where:

  • $O_i$: Observed frequency

  • $E_i$: Expected frequency (33.33 for each category)

Calculation Results of Chi-Square Test

Contributions by Category

Degrees of Freedom Calculation

• Degrees of Freedom ($df$): $3 - 1 = 2$

Conclusion of Chi-Square Test

Final Results and Implications

• Calculated Chai-Square ($ ext{χ}^2$): 26.66

• Critical Value ($ ext{α} = 0.05$): 5.99

• Result: Since 26.66 > 5.99, we reject the Null Hypothesis. The distribution of coffee preferences is not random.

• Note: The low ”No Preference” count indicates that consumers have strong opinions regarding these brands.

  • Note: Coffee was mentioned abstractly here; the initial context was about a sample preference inquiry.