Confidence Intervals for the Mean
Completion requirements
This section explains the need for confidence intervals and why a confidence interval is not the probability the interval contains the parameter. Then, it discusses how to compute a confidence interval on the mean when sigma is unknown and needs to be estimated. It also explains when to use t-distribution or a normal distribution. Next, it covers the difference between the shape of the t distribution and the normal distribution and how this difference is affected by degrees of freedom. Finally, it explains the procedure to compute a confidence interval on the difference between means.
t Distribution
Answers
- A t distribution has more scores in its tails and fewer in the center than a normal distribution, so it is leptokurtic. Also, a t distribution with 20 degrees of freedom has 95% of its distribution within 2.086 standard deviations of its mean.
- As the degrees of freedom increase, the t distribution looks more and more like a normal distribution.
- Use the "Find t for a confidence interval" calculator for a 90% confidence interval and you will get 1.753.
- Use the "t distribution" calculator and enter 10 for df and 2 for t. You will see that the probability of getting something greater than 2 SDs from the mean is .0734, so the probability of getting something less than 2 SDs from the mean is 1 - .0734 = .9266.
- There is a population of test scores with an unknown standard deviation. You sample 21 scores from this population, and you calculate the mean and standard deviation. You get a value for the mean that is 1.5 standard errors greater than what you think is the population mean. What is the probability that you would get a value 1.5 standard deviations or more from the mean in this t distribution? Answer: 0.148 or 14.8%