A confidence interval gives a range of possible values for a population parameter.
It main components are:

• Point estimate (\(\bar{x}\)) : best guess from the sample
  
• Confidence level (C) : how sure we are
  
• Margin of error (E) : how far off our estimate could be

Given population standard deviation \(\sigma\), the confidence interval is:

\[\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\] Where: - \(\bar{x}\) is the sample mean (point estimate) - \(z_{\alpha/2}\) is the critical z-value based on confidence level - \(\sigma\) is the population standard deviation - \(n\) is the sample size

• critical value for 90% is 1.645
• critical value for 95% is 1.96
• critical value for 99% is 2.576

Suppose there is a sample of 25 students with a mean test score of 75. The population standard deviation is 10.

Find the 95% confidence interval for the mean test score.

xbar = 75
sigma = 10
n = 25
criticalvalue = 1.96 
marginoferror = criticalvalue * (sigma / sqrt(n))
lowerbound = xbar - marginoferror
upperbound = xbar + marginoferror

• I am (whatever percent problem gave) confident that the (based on context of word problem, based on this problem it would be true mean test score) is between Lower Bound and Upper Bound

• Confidence Interval = Point Estimate ± Margin of Error
• Larger sample size means narrower intervals
• critical value for 90% is 1.645
• critical value for 95% is 1.96
• critical value for 99% is 2.576