Unit 7: Inference for Quantitative Data - Means

The t-Distribution

• Used when the population standard deviation is unknown, estimate using sample standard deviation (s).
• Introduced in 1908 by W. S. Gosset.
• For a normally distributed population, the t-distribution is bell-shaped and symmetric but more spread out than the normal distribution.
• Dependent on degrees of freedom (df), calculated as sample size minus one.
• Proper choice whenever population standard deviation is unknown, which is most real-world cases.

Confidence Interval for a Mean

• If sample size (n) is large:
  • Sample means are normally distributed.
  • Mean of sample means equals population mean (μ).
  • Standard deviation of sample means equals population standard deviation divided by square root of sample size.
• Use sample standard deviation when population standard deviation is unknown; called the standard error.
• Example 7.1: Gas mileage study with confidence intervals and significance testing.

Significance Test for a Mean

• Requires:
  • Simple random sample.
  • Sample size < 10% of population.
  • Large enough sample size for CLT or approximately normal distribution.
• Example 7.2: Testing manufacturer's claim about electricity usage with significance levels.

Confidence Interval for the Difference of Two Means

• Sampling distribution of differences of sample means is normally distributed.
• Use a t-distribution if population standard deviations are unknown.
• Example 7.3: Comparing accident rates between two departments with confidence intervals.

Significance Test for the Difference of Two Means

• Requires two independent simple random samples and either normally distributed populations or large enough sample sizes.
• Null hypothesis usually assumes equal means.
• Example 7.4: Comparing computer downtime between two companies with type I and type II errors.

Paired Data

• Involves one-sample analysis on differences from paired data.
• Example 7.5: SAT score improvements with confidence intervals.

Simulations and P-Values

• Simulations estimate P-values by modeling test statistic distributions under null hypothesis.
• Example 7.6: Machinery recalibration based on median absolute deviation calculations.

More on Power, Type I Errors, and Type II Errors

• Type II error: Failing to reject a false null hypothesis.
• Power: Probability of correctly rejecting a false null hypothesis.
• Example 7.7: Testing political support claims with power analysis.

Confidence Intervals Versus Hypothesis Tests

• Hypothesis tests assess claims about parameters; confidence intervals estimate parameters.
• Example 7.8: Comparing basketball player shooting accuracy with hypothesis tests and confidence intervals. Both decisions consistent across sample sizes.