Confidence Intervals
A confidence interval is a range of values that is likely to contain the population parameter with a specified level of confidence. The formula for a confidence interval for the mean (σ known) is x̄ ± Z_(α/2) × (σ/√n). When σ is unknown, the t-distribution is used: x̄ ± t_(α/2, n-1) × (s/√n). The confidence level (e.g., 95%) represents the proportion of intervals that would contain the true parameter if repeated samples were taken.
Knowledge Check
1.A sample of 36 students has a mean score of 75 with a population standard deviation of 12. What is the 95% confidence interval for the population mean? (Z = 1.96)
2.What is the critical Z-value for a 90% confidence interval?
3.When should the t-distribution be used instead of the normal distribution for confidence intervals?
4.As the sample size increases, the width of the confidence interval:
5.A sample has mean 50, standard deviation 10, and size 25. What is the 95% confidence interval using t-distribution? (t_0.025, 24 ≈ 2.064)
6.A 99% confidence interval is wider than a 95% confidence interval because:
7.What is the margin of error in a confidence interval?
8.A sample of 100 people has a mean height of 170 cm and standard deviation of 10 cm. What is the 95% confidence interval? (Z = 1.96)
9.The confidence level of 95% means that:
10.What is the critical t-value for a 95% confidence interval with degrees of freedom = 20? (Approximate)