1. CONFIDENCE INTERVALS
A confidence interval is an interval in which we are certain to a given probability of the mean lying in that interval.
For example if a 95% confidence interval is given, then 95% of the intervals would contain a true mean.
We used z-interval with normal distribution or CLT where the standard deviation of the population was known.
From a sample we usually don’t know anything about the population. So we use the t-distribution in theses situations and t-interval.
use sample mean and unbiased estimate of standard deviation.
As n increases, t - distribution becomes closer to normal distribution.

2. Examples how to use t-distribution.
To use t-distribution we need to know the sample size n.
The first step is to calculate the number of degrees of freedom
Example 2: If n =7, find the value of t
such that P(T<t)=0.90.
The GDC symbol for degrees of freedom is df.
Example 1: Find P(-2<T<4) if n=3.

3. Confidence intervals
Again if the standard deviation of the population is NOT known, and has to be estimated from the sample, then the distribution is altered to that of a Student-t distribution.
Example 1
Example 2 parts a and b only, without hypothesis testing yet.

4. Example 3 (given confidence interval, find confidence level).

9. (a) Determine unbiased estimates for µ and σ2.
Paper 3 Exam Question
Anna cycles to her new school. She records the times taken for the first ten days with the following results (in minutes).
Assume that these times are a random sample from the N(µ, σ2) distribution.
(a) Determine unbiased estimates for µ and σ2.
(2)
(b) Calculate a 95 % confidence interval for µ.
(3)
(a) estimate of μ = A1 estimate of σ2 = A1
(b) using a GDC (or otherwise), the 95% confidence interval is (M1) [12.6, 13.6] A1A1
Note: Accept open or closed intervals.