Confidence Intervals for Population Means: Summary Notes

Confidence Intervals for Population Means

• Purpose of Confidence Intervals: Used to estimate population means using sample data, accounting for variability.

• Constructing Confidence Intervals: Use sample mean $x$ to create an interval for population mean $μ$.

Critical Values for Confidence Intervals

• Critical Value $z^<em>$ or $t^</em>$: Determines the width of the confidence interval based on confidence level and sample size.

• One-Sample Z Interval: Used when population standard deviation $σ$ is known; formula: $x ext{±} z^* rac{σ}{ ext{√}n}$.

• One-Sample T Interval: Used when $σ$ is unknown; formula: $x ext{±} t^* rac{s}{ ext{√}n}$, where $s$ is the sample standard deviation.

Conditions for Confidence Intervals

• Random Sample: Data must come from a random sample of the population.

• 10% Condition: Sample size $n$ should be less than 10% of the population size $N$ when sampling without replacement.

• Normal/Large Sample Condition: Population should be normally distributed or sample size should be large ($n ext{≥} 30$).

Margin of Error and Sample Size

• Margin of Error (ME): Affected by sample size and confidence level; $ME ext{= } t^* rac{s}{ ext{√}n}$.

• Increasing sample size decreases margin of error (proportional to $rac{1}{ ext{√}n}$ ).

• Sample size can be estimated from previous studies or pilot studies, as $σ$ is often unknown.

• Example of Constructing Confidence Intervals: Determine mean values (e.g., books read, physical measurements) from samples, ensure conditions met, calculate using corresponding intervals.