Inferring Population Means and Comparing Groups: Confidence Intervals, Hypothesis Tests, and Variance Analysis

Confidence Intervals for Population Means

Introduction to Confidence Intervals

Confidence intervals are used to estimate population parameters by providing a range of plausible values, rather than a single point estimate. The confidence level represents the probability that the interval contains the true parameter.

• Point Estimate: The best single estimate of a parameter (e.g., sample mean).

• Confidence Interval: A range of values likely to contain the parameter, often written as $(\hat{y} - E, \hat{y} + E)$.

• Confidence Level: The probability (e.g., 95%) that the interval contains the parameter. Notation: $C$.

• Margin of Error: The distance between the point estimate and the bounds of the interval.

Example:

For a 95% confidence interval with point estimate $\hat{y} = 4$ and margin of error $E = 2$, the interval is $(2, 6)$.

Critical Values and Calculating Margin of Error

Critical values (z-scores or t-scores) are used to calculate the margin of error. The total area in the tails of a confidence interval is $\alpha = 1 - C$.

• Critical Value: $z_{\alpha/2}$ or $t_{\alpha/2}$, depending on whether the population standard deviation is known.

• Margin of Error Formula (z): $E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

• Margin of Error Formula (t): $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

Table: Common Critical Values

Confidence Level	z-value
90%	1.645
95%	1.960
99%	2.576

Population Standard Deviation Known vs. Unknown

When the population standard deviation ($\sigma$) is known, use the z-distribution. When unknown, use the t-distribution with sample standard deviation ($s$) and degrees of freedom ($df = n - 1$).

• z-distribution: Used when $\sigma$ is known and sample size $n > 30$ or population is normal.

• t-distribution: Used when $\sigma$ is unknown, sample size $n > 30$ or population is normal.

Example:

Construct a 99% confidence interval for a mean travel time with sample mean $\bar{x} = 1$ hour, $\sigma = 18$ minutes, $n = 36$.

Using TI-84 Calculator for Confidence Intervals

TI-84 calculators can compute confidence intervals using ZInterval or TInterval functions. Enter sample statistics or raw data, specify confidence level, and select the appropriate function.

• ZInterval: Use when $\sigma$ is known.

• TInterval: Use when $\sigma$ is unknown.

Minimum Sample Size for Confidence Intervals

Determining Sample Size

To achieve a desired margin of error, rearrange the margin of error formula to solve for sample size $n$.

• Sample Size Formula: $n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$

• Range Rule of Thumb: Estimate $\sigma$ as $\frac{\text{range}}{4}$ if not given.

Comparing Two Proportions

Hypothesis Tests for Two Proportions

To test claims about the difference between two proportions, use a pooled sample proportion and calculate the z-score.

• Null Hypothesis: $H_0: p_1 = p_2$

• Test Statistic: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\bar{p}\bar{q}}{n_1} + \frac{\bar{p}\bar{q}}{n_2}}}$

• Pooled Proportion: $\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}$

Confidence Intervals for Two Proportions

Construct a confidence interval for the difference in proportions using individual sample proportions.

• Margin of Error: $E = z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Interval: $((\hat{p}_1 - \hat{p}_2) - E, (\hat{p}_1 - \hat{p}_2) + E)$

Comparing Two Means

Hypothesis Tests for Two Means (Unknown, Unequal Variances)

When comparing means from two independent samples with unknown and unequal variances, use the t-distribution.

• Null Hypothesis: $H_0: \mu_1 = \mu_2$

• Test Statistic: $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom: $df = \min(n_1 - 1, n_2 - 1)$

Confidence Intervals for Two Means

Construct a confidence interval for the difference in means using the t-distribution.

• Margin of Error: $E = t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Interval: $((\bar{x}_1 - \bar{x}_2) - E, (\bar{x}_1 - \bar{x}_2) + E)$

Matched Pairs (Dependent Samples)

Introduction to Matched Pairs

Matched pairs occur when two data sets are related, and each value in one set is paired with a value in the other set. Common relationships include before-and-after measurements or related individuals.

• Difference Calculation: $d = x_1 - x_2$ for each pair.

• Mean Difference: $\bar{d}$

• Standard Deviation of Differences: $s_d$

Hypothesis Tests and Confidence Intervals for Matched Pairs

Replace the sample mean with the mean difference and use the t-distribution for hypothesis tests and confidence intervals.

• Test Statistic: $t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}$

• Confidence Interval: $((\bar{d} - E, \bar{d} + E))$ where $E = t_{\alpha/2} \cdot \frac{s_d}{\sqrt{n}}$

Hypothesis Test for Correlation Coefficient

Testing the Significance of Correlation

The correlation coefficient ($r$) measures the strength and direction of a linear relationship. Hypothesis tests can determine if the correlation is statistically significant.

• Null Hypothesis: $H_0: \rho = 0$ (no correlation)

• Alternative Hypothesis: $H_a: \rho \neq 0$ (correlation exists)

• Test Statistic: Use LinRegTTest on TI-84.

F-Distribution and Two Variances

Introduction to F-Distribution

The F-distribution is used to compare two variances. It is always right-skewed and takes only positive values. The F-statistic is calculated as the ratio of two sample variances.

• Null Hypothesis: $H_0: \sigma_1^2 = \sigma_2^2$

• Test Statistic: $F = \frac{s_1^2}{s_2^2}$

• Degrees of Freedom: $df_1 = n_1 - 1$, $df_2 = n_2 - 1$

Hypothesis Tests for Two Variances Using TI-84

Use the 2-SampFTest function on TI-84 to perform hypothesis tests for two variances.

Summary Table: Key Formulas

Concept	Formula
Margin of Error (z)	$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
Margin of Error (t)	$E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
Sample Size	$n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$
Range Rule of Thumb	$s \approx \frac{\text{range}}{4}$
Degrees of Freedom	$df = n - 1$

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