Stat 211: Elementary Inferential Statistics - Unit 6 Study Notes

Stat 211: Elementary Inferential Statistics

Unit 6: One-Sample Inference

Confidence Intervals for Proportions

• Example: 2024 Births

  • In 2024, 3,622,673 babies were born in the US.

  • The proportion of premature births in 2024 was 10.4%.

Sampling Distribution

• Definition: The sampling distribution for proportions specifies how sample proportions vary.

• One individual proportion is derived from a sample of 100 babies.

  • Distribution Characteristics:

  • Normally distributed with mean: $p = 0.104$

  • Standard deviation determined as:
    $ext{Standard deviation} = rac{p(1-p)}{n}$

  • Specific normal distribution:
    $N(0.104, 0.031)$

• Sample Numbers:

  • # of Samples: 150, 100, 50, 50.

Recall: Empirical Rule

• This rule allows for probability estimations from a normally distributed dataset:

  • 0.15% | 2.35% | 13.5% | 34% | 34% | 68% of values fall within:

  • 1 (68%)

  • 2 (95%)

  • 3 (99.7%) standard deviations from the mean.

• Application: Empirical rules extend to estimating confidence intervals.

Our Goal

• Objective: Estimate population parameters from a sample

• Single values lack precision; for example, if a sample of 100 babies has 14 preemies, $p = 0.14$.

• Computing standard deviation from just this sample is not ideal.

  • Standard Error (SE):

  • Definition: Estimated standard deviation of a sampling distribution

  • Significance: Estimates variability of sample statistics (p or x).

  • Formula for SE of Sample Proportions:
    $SE(p) = rac{p(1-p)}{n}$

  • Applying the SE Calculation:

    • For $p = 0.14$:
      $SE(p) = rac{0.14(1-0.14)}{100} <br>ightarrow SE(p) = 0.035$

Rationale for Confidence Intervals

• We utilize sample statistics and account for uncertainty using the standard error.

  • Estimated sampling distribution is:

  • From sample: $N(0.14, 0.035)$

  • True distribution: $N(0.104, 0.031)$

• Confidence Interval Interpretation:

  • If 95% of similar samples lay between 7% and 21%, then:

  • 0.07 < p < 0.21 or (7%, 21%) indicates we are 95% confident the true population proportion is within this interval.

Rough Confidence Interval

• A rough approximation for 95% CI is:

  • $ext{CI} <br>ightarrow p ext{ ± } 2( ext{Standard Deviation})$

• Critical z-Values: Used to define confidence levels, calculated from standard normal distribution:

  • Computation Examples:

  • Confidence Level 90% → Critical z-Value ≈ 1.645

  • Confidence Level 95% → Critical z-Value ≈ 1.960

  • Confidence Level 99% → Critical z-Value ≈ 2.576

More Precise Confidence Interval

• Apply more precise definition:

  • $ext{CI} <br>ightarrow p ± 1.96( ext{SE})$

  • Example for $p = 0.14$:

  • $0.14 ± 1.96(0.035)$

• This is known as the one-proportion z-interval.

Misinterpretations in Confidence Intervals

• Commonly misphrased interpretations include:

  • "14% of all babies born in 1998 were born prematurely."

  • "It is probably true that 14% of all babies born in 1998 were born prematurely."

  • Incorrectly stating that the true proportion is within a computed interval is misleading.

• What to Say Instead:

  • Statements like "We are 95% confident the true population proportion lies within this interval" are more accurate.

One-Proportion z-interval

• Conditions required to find a confidence interval include:

  • Previous formulas apply with ED(p) defined as:

  • $p ± z* rac{p(1 − p)}{n}$

  • Where z* specifies the number of SEs needed for C% of random samples.

Why Confidence?

• Quantifying uncertainty is crucial; confidence intervals provide clarity beyond simple point estimates.

  • “Confidence” elucidates the long-run success of the method over multiple samples.

  • The value of confidence intervals is in their broader representativeness, not the accuracy of any single interval determination.

Example: Pineapple on Pizza Survey

• From a 2025 survey, 1,000 adults revealed:

  • 591 people who preferred pineapple on pizza.

• Calculate the 95% confidence interval for the population proportion:

  • Using the confidence interval applet, indications for parameters include:

  • Number of Successes: 591

  • Total Sample Size: 1000

  • Confidence Level: 95%

  • Output from Calculator:

  • Confidence Interval: (0.5605, 0.6215)

Example: Boba Tea

• Aim: Compute a 90% confidence interval for the proportion of college students who tried boba tea.

  • The interval is determined as:

  • Point Estimate ± z*•Standard Error.

Confidence vs. Precision

• Margin of Error Explanation:

  • Tradeoff between confidence level and interval width:

  • More confidence → Wider interval.

  • Less confidence → Narrower interval.

• Example intervals represent confidence vs. precision:

  • 50% CI: (0.09, 0.11) yields smaller margin of error

  • 100% CI: [0, 1] indicates maximal confidence with uncertainty.

Election Polling and Confidence Intervals

• Polling samples report proportions ± margin of error:

  • E.g., Candidate A at 52% with ±3% ME gives CI: [49%, 55%].

• Importance of sample size calculations for determining required sample sizes to achieve desired confidence levels and margin of error.

Choosing a Sample Size

• Estimation methods differing based on prior estimates:

  • When p is known: $n = rac{z^2 imes p(1-p)}{ME^2}$

  • When p is not known (use conservative): $n = rac{z^2}{2 ME^2}$

• Always round up in sample size calculations.

• Example Calculations:

  • For 3% margin of error with 95% confidence, determine appropriate sample size.

Confidence Intervals for Means

• Example: 2024 Birthweights

  • In 2024, average birthweight was 3,318.9 grams.

• Sampling Distribution:

  • Central Limit Theorem applies; distribution generated by sample means can be modeled as normal:

  • ar{x} ext{ drawn from } N( ext{Population Mean } ext{μ}, ext{ Population SD } σ)

  • Mean calculated:

  • $N(3318.9, 60.95)$

Confidence Interval for Mean (σ Known)

• Formula for computing confidence interval:

  • ext{CI} = ar{x} ± z^* rac{σ}{ ext{sqrt}(n)}.

Example: Birthweights for Confidence Interval

• Most commonly encounter unknown population standard deviation (use sample standard deviation:.

• t-Distribution Introduction:

  • Derived by William S. Gosset under pseudonym