Confidence intervals help estimate population means using sample data. They're crucial when we know the population's standard deviation or have a large sample size. The Central Limit Theorem allows us to create these intervals, giving us a range of likely values for the true population mean.

Sample size and confidence level impact interval width. Larger samples give more precise estimates, while higher confidence levels widen intervals. Researchers must balance precision and certainty when choosing confidence levels, considering the tradeoffs between narrow and wide intervals.

Confidence Intervals for Population Mean with Known Standard Deviation or Large Sample Size

Confidence intervals using Central Limit Theorem

• Central Limit Theorem enables creating confidence intervals for population mean ($\mu$) when population standard deviation ($\sigma$) is known or sample size is large ($n \geq 30$)
• Confidence interval formula: $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
  • $\bar{x}$ represents sample mean (point estimate)
  • $z_{\alpha/2}$ represents critical value from standard normal distribution based on desired confidence level
  • $\sigma$ represents known population standard deviation
  • $n$ represents sample size
• Find critical value ($z_{\alpha/2}$) using standard normal distribution table or calculator
  • 95% confidence level: $\alpha = 0.05$ and $z_{\alpha/2} = 1.96$
  • 99% confidence level: $\alpha = 0.01$ and $z_{\alpha/2} = 2.58$
• Confidence interval provides range of plausible values for population mean based on sample data
• Example: Estimating average height of students in a school with known standard deviation of 5 cm and sample mean of 170 cm ($n = 50$) at 95% confidence level
  • $170 \pm 1.96 \cdot \frac{5}{\sqrt{50}} = (168.6, 171.4)$

Effects of sample size on intervals

• Sample size ($n$) impacts width of confidence interval
  • Increasing sample size decreases width of confidence interval
  • Larger sample sizes yield more precise estimates of population mean
• Example: Doubling sample size from 50 to 100 students in height example
  • $170 \pm 1.96 \cdot \frac{5}{\sqrt{100}} = (169.0, 171.0)$
  • Interval width reduced from 2.8 cm to 2.0 cm
• Margin of error quantifies range of values added and subtracted from sample mean to create confidence interval
  • Margin of error formula: $z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
  • Increasing sample size or decreasing confidence level reduces margin of error, resulting in narrower interval

Tradeoffs in confidence level vs width

• Inverse relationship exists between confidence level and interval width
  • Increasing confidence level widens interval, while decreasing confidence level narrows interval
• Higher confidence levels (99%) provide more certainty that true population mean falls within interval
  • Increased certainty comes at cost of wider, less precise interval
• Lower confidence levels (90%) result in narrower intervals, providing more precise estimate of population mean
  • Increased precision comes with lower confidence that true population mean falls within interval
• Researchers must balance desired confidence level with need for precision when selecting confidence level for study
• Example: Comparing 90%, 95%, and 99% confidence intervals for student height example ($n = 50$)
  • 90% CI: $170 \pm 1.65 \cdot \frac{5}{\sqrt{50}} = (168.8, 171.2)$
  • 95% CI: $170 \pm 1.96 \cdot \frac{5}{\sqrt{50}} = (168.6, 171.4)$
  • 99% CI: $170 \pm 2.58 \cdot \frac{5}{\sqrt{50}} = (168.2, 171.8)$

Statistical Concepts and Confidence Intervals

• Sampling distribution forms the basis for constructing confidence intervals
• Z-score measures the number of standard deviations a data point is from the mean
• Hypothesis testing uses confidence intervals to make inferences about population parameters
• Degrees of freedom affect the shape of the sampling distribution for small sample sizes