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 values in which the population value is likely to occur Usually given in the format: Estimate ( 𝑥 𝑜𝑟 𝑝 ) ± 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 𝜎 for the population we can now find the confidence interval using a calculated sample mean ( ). IQ score example: For, 𝜇 𝑥 =112 𝑎𝑛𝑑 𝜎=15 , what would be the confidence interval at an 80% confident level?

Interpretations of Confidence Intervals and Confidence Levels This means that we are 80% confident that the TRUE mean IQ score of ALL freshmen at Big City University falls between 109.28 and 114.72. Confidence Level: Being “80% confident” means that 80% of the time, repeated samples of the same size will produce an interval that contains the TRUE mean.

Make the s value smaller Make the n (sample size) larger Often a high confidence level (95 or 99%), means that your interval must be very large (high margin of error). Ultimately, we would like to create a confidence interval with a high confidence level and very small margin of error. How can we control that??? Make the z* value smaller that causes you to accept a lower confidence level. Make the s value smaller this does make it easier to get a more accurate m, but is difficult to do Make the n (sample size) larger dividing by a larger number makes smaller and in turn the margin of error smaller Best Option!

So how do we find the desired sample size? Let m = the desired margin of error So, Now solve for n. Pg 640 #20b   The sample size must be at least 2653 people