Point and Interval Estimation

• Estimation determines likely values for a population parameter based on a sample

Types of Estimates

• Point Estimate: A single value estimating the population parameter
• Interval Estimate: A range of values where the population parameter is likely to fall

Point Estimation

• A single number estimates an unknown population parameter
• The sample mean $(\bar{x})$ is a point estimate of the population mean $(\mu)$
• Example: Average height of 100 students is 175 cm, so the point estimate for the school's average height is 175 cm

Interval Estimation

• Gives a range where the population parameter is believed to lie
• Associated with a confidence level, indicating the probability the interval contains the true parameter
• Example: A 95% confidence interval for average student height might be (170 cm, 180 cm)

Confidence Interval

• An interval estimate of a population parameter
• Associated with a confidence level that indicates the probability that the interval contains the true parameter
• Confidence Interval = Point Estimate ± (Critical Value × Standard Error)

Key Components

• Point Estimate: Sample statistic estimates the population parameter
• Critical Value: From standard distribution (Z or t) and determines the width of the confidence interval
• Standard Error: Measures variability of the sample statistic

Confidence Level

• Probability that the confidence interval contains the true population parameter
• Common levels: 90%, 95%, 99%

Margin of Error

• The amount added/subtracted from the point estimate to form the confidence interval
• Margin of Error= Critical Value × Standard Error

Z-Interval

• Use when the population standard deviation ($\sigma$) is known, the sample size is large (> 30), and data is normally distributed
• Formula: $\bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
  • $\bar{x}$ = Sample mean
  • $Z_{\alpha/2}$ = Z-score for the desired confidence level
  • $\sigma$ = Population standard deviation
  • n = Sample size

T-Interval

• Use when the population standard deviation ($\sigma$) is unknown, the sample size is small (< 30), and data is normally distributed
• Formula: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
  • $\bar{x}$ = Sample mean
  • $t_{\alpha/2, n-1}$ = t-score for desired confidence level and degrees of freedom
  • s = Sample standard deviation
  • n = Sample size

Example 1: Z-Interval

• A survey of 500 adults found 60% prefer coffee. Assuming a population standard deviation of 5%, a 95% confidence interval for the proportion of adults who prefer coffee is (0.556, 0.644)

Example 2: T-Interval

• A sample of 25 students has a mean test score of 75 with a sample standard deviation of 8. A 90% confidence interval for the population mean test score is (72.26, 77.74)

Key Points

• Estimation uses sample data to infer population parameters
• Point estimates are single values, while interval estimates offer a range
• Confidence intervals quantify the uncertainty in estimating population parameters
• Choosing between Z and T-Intervals depends on knowing the population standard deviation and the sample size