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 .

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

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

Example:

For a 95% confidence interval with point estimate and margin of error , the interval is .

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 .

• Critical Value: or , depending on whether the population standard deviation is known.

• Margin of Error Formula (z):

• Margin of Error Formula (t):

Table: Common Critical Values

Population Standard Deviation Known vs. Unknown

When the population standard deviation () is known, use the z-distribution. When unknown, use the t-distribution with sample standard deviation () and degrees of freedom ().

• z-distribution: Used when is known and sample size or population is normal.

• t-distribution: Used when is unknown, sample size or population is normal.

Example:

Construct a 99% confidence interval for a mean travel time with sample mean hour, minutes, .

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 is known.

• TInterval: Use when 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 .

• Sample Size Formula:

• Range Rule of Thumb: Estimate as 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:

• Test Statistic:

• Pooled Proportion:

Confidence Intervals for Two Proportions

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

• Margin of Error:

• Interval:

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:

• Test Statistic:

• Degrees of Freedom:

Confidence Intervals for Two Means

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

• Margin of Error:

• Interval:

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: for each pair.

• Mean Difference:

• Standard Deviation of Differences:

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:

• Confidence Interval: where

Hypothesis Test for Correlation Coefficient

Testing the Significance of Correlation

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

• Null Hypothesis: (no correlation)

• Alternative Hypothesis: (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:

• Test Statistic:

• Degrees of Freedom: ,

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

Additional info: These notes expand on the original content by providing definitions, formulas, and examples for each major topic, ensuring completeness and academic quality for exam preparation.