Inference: Conclusion with Confidence Strand IV Inference: Conclusion with Confidence Statistical Inference: drawing conclusions about a population based upon the data collected from a sample of that population. To determine if our conclusion are correct/reasonable, we can use the concepts of probabilities to express the strength of our conclusions. In chapter 10, we will discuss confidence intervals which will allow us to estimate the value of a population parameter.

Section 10.1: Confidence Intervals A range of population values in which our sample value is likely to occur Usually given in the format: estimate( ) ± margin of error or (estimate – margin of error, estimate + margin of error) Confidence Level: the probability that the interval will capture the true parameter value in repeated samples (the probability that our estimation method is successful)

Example: The Sampling Distribution of the mean score ( ) of an SRS of 50 Big City University freshmen on an IQ test If we wanted to be 95% confident that we could estimate the mean IQ score for the population, we would use a margin of error equal to approximately 2 standard deviations from the estimated mean.

Conditions for Constructing a Confidence Interval for m These conditions MUST be checked and validated first… SRS: the data came from a simple random sample of the population of interest Normality: the sampling distribution of is approximately normal (Use CLT to prove this ) Independence: individual observations are independent when sampling without replacement and population ≥ 10 * sample size This also assumes that you know the value of s.

What would the critical values for 90%, 95%, and 99% be? Investigation: Let’s say that you want to create a confidence interval with an 80% confidence level for a SRS that is approximately normal and independent. Using z-scores can help us to find a confidence interval about the unknown mean (m). What would those z-score values be? z-score values that are used to create cutoff values for a confidence interval are called critical values (the positive value is represented as z*) What would the critical values for 90%, 95%, and 99% be?

The critical values are the number of standard deviations from the mean that will give you the confidence level you want. Since we know the for the population we can now find the confidence interval using a calculated sample mean ( ). IQ score example: = 112 and s = 15, What would be the confidence interval at an 80% confident level? (109.28 , 114.72), we are 80% confident that the average IQ score for all Big City University freshmen would fall between 109.28 and 114.72.