Statistical Inference Statistical Inference is the process of making judgments about a population based on properties of the sample Statistical Inference is the process of making judgments about a population based on properties of the sample intuition only goes so far towards making decisions of this nature intuition only goes so far towards making decisions of this nature experts can offer conflicting opinions using the same data experts can offer conflicting opinions using the same data

Methods of Statistical Inference Estimation Estimation Predict the value of an unknown parameter with specified confidence Predict the value of an unknown parameter with specified confidence Decision Making Decision Making Decide between opposing statements about the population (parameter) Decide between opposing statements about the population (parameter)

Estimation Estimating a Population Mean (  ) Estimating a Population Mean (  ) Point Estimate Point Estimate Mean, median, mode, etc.Mean, median, mode, etc. Easy to calculate and use, but random in valueEasy to calculate and use, but random in value Interval Estimate Interval Estimate Range of values containing parameterRange of values containing parameter Unknown accuracy within rangeUnknown accuracy within range Confidence Interval Confidence Interval Interval with known probability of containing truthInterval with known probability of containing truth Often based on a “pivot statistic” with known distributionOften based on a “pivot statistic” with known distribution

Estimating  known  The central limit theorem provides a sampling distribution for the sample mean in cases of sufficient sample size (n ≥ 30). The following probability statement can be used to find a confidence interval for μ: The central limit theorem provides a sampling distribution for the sample mean in cases of sufficient sample size (n ≥ 30). The following probability statement can be used to find a confidence interval for μ:

(1-  ) Confidence Interval for μ An alternative form is: An alternative form is:

Confidence Intervals The level of confidence and sample size both effect the width of the confidence interval. The level of confidence and sample size both effect the width of the confidence interval. Increasing the level of confidence results in a wider confidence interval. Increasing the level of confidence results in a wider confidence interval. Increasing the sample size results in a narrower confidence interval. Increasing the sample size results in a narrower confidence interval. Setting the level of confidence too high results in a confidence interval that is too wide to be of any practical use. Setting the level of confidence too high results in a confidence interval that is too wide to be of any practical use. i.e. 100% confidence intervals are from -∞ to ∞i.e. 100% confidence intervals are from -∞ to ∞

Bootstrap Confidence Intervals The bootstrap technique can be used to obtain a confidence interval estimate. The bootstrap technique can be used to obtain a confidence interval estimate. Simulate 1000 bootstrap samples from the data. Simulate 1000 bootstrap samples from the data. The 25 th order statistic and the 975 th order statistic are used as the lower and upper bounds, respectively. The 25 th order statistic and the 975 th order statistic are used as the lower and upper bounds, respectively. This is a non-parametric approach since no assumptions are made about the underlying distribution of the data. This is a non-parametric approach since no assumptions are made about the underlying distribution of the data.

Necessary Sample Size for Estimating the Mean (  The sample size necessary to estimate the mean (μ) with a margin of error E and (1-α) level of confidence is: The sample size necessary to estimate the mean (μ) with a margin of error E and (1-α) level of confidence is:

What if  is  unknown  William Gossett - a chemist for Guiness Brewery in the early 1900's discovered that substituting s for σ in the margin of error formula, William Gossett - a chemist for Guiness Brewery in the early 1900's discovered that substituting s for σ in the margin of error formula, resulted in a confidence interval that was too narrow for the desired level of confidence (1- α). Resulted in increased error rate in statistical inference. Resulted in increased error rate in statistical inference. Error rate particularly noticeable for small samples Error rate particularly noticeable for small samples

Student t Distribution Gossett discovered that the statistic, Gossett discovered that the statistic, has a Student t distribution with degree of freedom equal to n-1. The t distribution: is symmetric about 0 is symmetric about 0 has heavier tails than the normal distribution has heavier tails than the normal distribution converges to the normal distribution as n  ∞. converges to the normal distribution as n  ∞.

Estimating  known  If the underlying data is from a normal distribution and the standard deviation is unknown, then the probability statement can be used to find a confidence interval for μ: If the underlying data is from a normal distribution and the standard deviation is unknown, then the probability statement can be used to find a confidence interval for μ:

Estimating  unknown  If the sample data is from a normal distribution and  is unknown, then the (1-  ) Confidence Interval for μ is: If the sample data is from a normal distribution and  is unknown, then the (1-  ) Confidence Interval for μ is:

Why settle for small sample size? Can’t you just collect more data? Can’t you just collect more data? Samples can be expensive to obtain. Samples can be expensive to obtain. shuttle launch, batch runshuttle launch, batch run Samples can be difficult to obtain. Samples can be difficult to obtain. rare specimen, chemical processrare specimen, chemical process Samples can be time consuming to obtain. Samples can be time consuming to obtain. cancer research, effects of timecancer research, effects of time Ethical questions can arise. Ethical questions can arise. medical research can't continue if initial results look badmedical research can't continue if initial results look bad

Estimating Population Proportion (p) Can be thought of as the binomial probability of success if randomly sampling from the population. Can be thought of as the binomial probability of success if randomly sampling from the population. Let p be the proportion of the population with some characteristic of interest. The characteristic is either present or it is not present, so the number with the characteristic is binomial. Let p be the proportion of the population with some characteristic of interest. The characteristic is either present or it is not present, so the number with the characteristic is binomial.

Estimating Population Proportion (p) The central limit theorem applies to a binomial random variable with sufficient sample size The central limit theorem applies to a binomial random variable with sufficient sample size Expected number of successes (n · p) and failures (n · q) must be at least 5. Expected number of successes (n · p) and failures (n · q) must be at least 5. The number of successes (X) is normally distributed with mean n · p and variance n · p · q. The number of successes (X) is normally distributed with mean n · p and variance n · p · q. The proportion of interest is normally distributed, with mean p and variance of p · q/n. The proportion of interest is normally distributed, with mean p and variance of p · q/n.

(1-  ) Confidence Interval for a Population Proportion (p) The point estimate for proportion is: The point estimate for proportion is: The (1-  ) Confidence Level for p is: The (1-  ) Confidence Level for p is:

Necessary Sample Size for Estimating the Proportion (p  The sample size necessary to estimate the proportion (p) a margin of error E and (1-  ) level of confidence (a) with prior knowledge of p and q and (b) no prior knowledge of p and q is. The sample size necessary to estimate the proportion (p) a margin of error E and (1-  ) level of confidence (a) with prior knowledge of p and q and (b) no prior knowledge of p and q is.

Estimating the Population Variance  Point Estimate of  2 :

Χ 2 Distribution The statistic, The statistic, has a Chi-Square distribution with degree of freedom equal to n-1. This distribution is skewed right and converges to the normal distribution as n  ∞.

Estimating   The following probability statement can be used to find a confidence interval for   : The following probability statement can be used to find a confidence interval for   :

(1-  ) Confidence Interval for a Population Variance The (1-  ) Confidence Level for  2 is: The (1-  ) Confidence Level for  2 is: