Confidence intervals for a mean (the impact of not knowing the standard deviation)

The applet below allows one to visually investigate confidence intervals for a population mean. Specify the sample size n, the shape of the distribution (Normal or Right skewed), the true population mean (Mean) and the true population standard deviation (Std. Dev.). When you click the Simulate button, 100 separate samples of size n will be selected from a population with these population parameters. For each of the 100 samples, a 95% Z confidence interval (known standard deviation) and a 95% T confidence interval (unknown standard deviation) are displayed in the plot to the right. If an interval does not contain the true mean, it is displayed in red. Additional simulations can be carried out by clicking the Simulate button multiple times. The cumulative number of times that each type of interval contains the true mean is also tabled. Press the Clear button to clear existing results and start a new simulation. Things to try with the applet:

  • Simulate at least 1000 intervals using samples of size 10 from a normal distribution with a Mean = 50 and Std. Dev. = 1. What proportion of the 95% Z confidence intervals contain 50? What proportion of the 95% T confidence intervals contain 50?
  • Simulate at least 1000 intervals using samples of size 100 from a normal distribution with a Mean = 50 and Std. Dev. = 1. What proportion of the 95% Z confidence intervals contain 50? What proportion of the 95% T confidence intervals contain 50?
  • Simulate at least 1000 intervals using samples of size 10 from a right skewed distribution with a Mean = 50 and Std. Dev. = 1. What proportion of the 95% Z confidence intervals contain 50? What proportion of the 95% T confidence intervals contain 50?
  • Simulate at least 1000 intervals using samples of size 100 from a right skewed distribution with a Mean = 50 and Std. Dev. = 1. What proportion of the 95% Z confidence intervals contain 50? What proportion of the 95% T confidence intervals contain 50?