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.

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Presentation transcript:

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.

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.

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 https://osc-ib.com/ib-video-revision/confidence-interval Example 2 parts a and b only, without hypothesis testing yet. https://osc-ib.com/ib-video-revision/confidence-intervals

Example 3 (given confidence interval, find confidence level). https://osc-ib.com/ib-video-revision/confidence-levels

(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). 12.4 13.7 12.5 13.4 13.8 12.3 14.0 12.8 12.6 13.5 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 = 13.1 A1 estimate of σ2 = 0.416 A1 (b) using a GDC (or otherwise), the 95% confidence interval is (M1) [12.6, 13.6] A1A1 Note: Accept open or closed intervals.