Inferences about Means

T Distribution and Sampling Distribution Models for Means

Introduction

• The lecture focuses on inferences about means, specifically confidence intervals, using the t-distribution.

Standard Error and Margin of Error

• Standard Error: $\frac{s}{\sqrt{n}}$ (where $s$ is the sample standard deviation and $n$ is the sample size).
• Margin of Error: $t^* \frac{s}{\sqrt{n}}$ (where $t^*$ is the critical t-value).

The Importance of Sample Size

• Larger samples are better.
• Example: Corned beef weights sold to ShopRite have a mean of 3.9 lbs and a standard deviation of 1.2 lbs.
  • A case contains 9 packages.
  • A gross contains 144 packages.
  • Standard deviation of the mean weight in a case: $\frac{1.2}{\sqrt{9}} = 0.4$
  • Standard deviation of the mean weight in a gross: $\frac{1.2}{\sqrt{144}} = 0.1$

Gosset’s t-Distribution

• William S. Gosset, working for Guinness Brewery, developed the Student's t-distribution.
• The t-models are a family of distributions that depend on degrees of freedom (df), denoted as $t_{df}$.

Properties of t-Distributions

• As degrees of freedom increase, t-models approach the Normal distribution.
• A t-model with infinite degrees of freedom is equivalent to the Normal distribution.
• t-models are unimodal, symmetric, and bell-shaped, similar to the Normal distribution.
• t-distributions have heavier tails than the Normal distribution, especially with small degrees of freedom, leading to larger margins of error.

Using Table T and Calculators

• Use invT function on calculators to find $t^<em>$ values, similar to using invNorm for $z^</em>$.
• area = $\frac{(1-C)}{2}$, where C is the confidence level.
• df = n - 1

Examples - Degrees of Freedom and t* Values

• Examples provided for finding degrees of freedom (df) and corresponding $t^*$ values.
• df = 19, $t^*$ = 2.093
• df = 19, $t^*$ = 1.729
• df = 11, $t^*$ = 3.106
• df = 39, $t^*$ = 2.023

Steps for Creating a Confidence Interval

1. Conditions.
2. Calculations/Mechanics.
3. Conclusion in Context.

Assumptions and Conditions

• Independence Assumption:
  • Randomization Condition: Data from a random sample or randomized experiment.
  • 10% Condition: Sample size should be no more than 10% of the population when sampling without replacement.
• Nearly Normal Condition:
  • For small sample sizes (n < 15), data should closely follow a Normal model, checked with a histogram.
  • For moderate sample sizes (15 < n < 30), t-distribution works if data is unimodal, symmetric, and has no outliers.
  • For large sample sizes (n > 30), t-methods are generally safe unless data is extremely skewed.

Visualizing Data

• Use pictures to understand data better.
• Check Nearly Normal Condition with graphs, such as histograms, to verify unimodality, symmetry, and absence of outliers.

Formula for One-Sample t-Interval for the Mean

• Confidence Interval formual: $\bar{x} \pm t^*_{n-1} \frac{s}{\sqrt{n}}$
• Can also be found by using the t-interval function on calculators.

Example: Battery Lifespan

• A company wants to develop a battery lasting over 300 minutes.
• A test of 12 batteries yielded lifespans (in minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326, 311, 288.
• Find a 90% confidence interval for the mean lifespan.
• Check Conditions: 12 batteries are less than 10% of all batteries.

Sample Size Determination

• To find the sample size needed for a specific confidence level and margin of error (ME), solve for n:
  • $ME = t^* \frac{s}{\sqrt{n}}$
• Challenges:
  • Values are often unknown.
  • Solutions:
    • Use $s$ from a pilot study.
    • Use $z^<em>$ in place of $t^</em>$.

Important Considerations for Sample Size

• Sample size calculations are approximate.
• The margin of error after data collection may not match the planned ME.
• The sample size formula relies on quantities unknown until data collection.
• Ensure the sample size is sufficient to provide meaningful insights.

Example on Calculating Sample Size

• Follow up to the batteries example.

What Can Go Wrong?

• Avoid confusing proportions and means.
• Be cautious of multimodality and skewed data.
  • For skewed data, consider re-expressing the variable.
• Set outliers aside and report them separately.
• Correctly interpret the confidence interval; it's about the population mean, not sample means or individuals.

Review of Key Concepts

• Estimating standard deviation with standard error: $SE = \frac{s}{\sqrt{n}}$
• Using Student’s t-model with n – 1 degrees of freedom to account for uncertainty from using the SE.

AP Tips

• Check conditions carefully.
• Draw a graph on paper when checking a graph for a small sample procedure.
• If only summary statistics are provided, the sample size is likely large enough for the CLT to apply.

Means - Simulation with Dice

• Demonstration of the Central Limit Theorem using simulations of dice tosses.
• 10,000 tosses of one die.
• Averaging two, three, five, and twenty dice.
• As the number of dice increases, the distribution of the average approaches a Normal distribution, tightening around the population mean (3.5).

Central Limit Theorem (CLT)

• The sampling distribution of any mean becomes more nearly Normal as the sample size grows.
• Conditions: independent observations collected with randomization.
• The CLT applies regardless of the shape of the population distribution.

Variation and Sample Size

• The standard deviation of the sampling distribution decreases with the square root of the sample size.
• Variability decreases as sample size increases.
• The square root limits the extent to which larger samples improve information about the population (Law of Diminishing Returns).