Probability = Relative Frequency

Typical Distribution for a Discrete Variable

Typical Distribution for a Continuous Variable Normal (Gaussian) Distribution

Probability Density Function

The Normal Distribution (Also termed the “Gaussian Distribution”) Note: f(x)dx is the probability of observing a value of x between x and x+dx.

Selecting One Normal Distribution The Normal Distribution can fit data with any mean and any standard deviation…..which one shall we focus on? We do need to focus on just one….for tables and for theoretical developments.

Need for the Standard Normal Distribution The mean, , and standard deviation, , depends upon the data----a wide variety of values are possible To generalize about data we need: – to define a standard curve and –a method of converting any Normal curve to the standard Normal curve

The Standard Normal Distribution  = 0  = 1

The Standard Normal Distribution

P.D.F. of z

Transforming Normal to Standard Normal Distributions Observations x i are transformed to z i : This allows us to go from f(x) versus x to f(z) versus z. Areas under f(z) versus z are tabulated.

The Use of Standard Normal Curves Statistical Tables Convert x to z Use tables of area of curve segments between different z values on the standard normal curve to define probabilities

Emphasis on Mean Values We are really not interested in individual observations as much as we are in the mean value. Now we have f(x) versus x where x is the value of observations. We need to deal in xbar, the sample mean, instead of individual x values.

Introduction to Inferential Statistics Inferential statistics refers to methods for making generalizations about populations on the basis of data from samples

Sample Quantities Mean Standard Deviation is an estimate of  is an estimate of  Note: These quantities can be for any distribution, Normal or otherwise.

Population and Sample Measures Parameters: Mean of the Population   Standard Deviation of the Population   Variance of the Population   2 Statistics (sample estimates of the parameters): Sample estimate of   Sample Estimate of   s

Population and Samples n observations per sample.

P.D.F. of the Sample Means Note: The std. dev. of this distribution is  xbar

Types of Estimators Point estimator - gives a single value as an estimate of the parameter of interest Interval estimator - specifies a range of values of the parameter and our confidence that the parameter value is in that range

Point Estimators Unbiased estimator: as the number of observations, n, increases for the sample the average value of the estimator approaches the value of the population parameter.

Interval Estimators P(lower limit<parameter<upper limit) =1-  lower limit and upper limit = confidence limits upper limit-lower limit=confidence interval 1-  = confidence level; degree of confidence; confidence coefficient

Comments on the Need to Transform to z for C.I. of Means We have a point estimate of , xbar. Now the interval estimate consists of a lower and an upper bound around our point estimate of the population mean: P(  low <  <  high )=1- 

Confidence Interval for a Population Mean P(  low <  <  high )=1-  If f(xbar) versus xbar is a Normal distribution and if we can define z as we did before, then:  low =xbar-z  /2  xbar  high =xbar+z  /2  xbar

A Standard Distribution for f(xbar) versus xbar Previously we transformed f(x) versus x to f(z) versus z We can still use f(z) versus z as our standard distribution. Now we need to transform f(xbar) versus xbar to f(z) versus z.

P.D.F. of the Sample Means Note: The std. dev. of this distribution is  xbar

P.D.F. of z

Transforming Normal to Standard Normal Distributions This time the sample means, xbar are transformed to z: Note that now we use xbar and sigma for the p.d.f. of xbar.

The Normal Distribution Family

Remaining Questions When can we assume that f(xbar) versus xbar is a Normal Distribution? –when f(x) versus x is a Normal Distribution –but….what if f(x) versus x is not a Normal Distribution How can we calculate μ and σ for the f(xbar) versus xbar distribution?

The Answer to Both Questions The Central Limit Theorem

If x is distributed with mean  and standard deviation , then the sample mean (xbar) obtained from a random sample of size n will have a distribution that approaches a normal distribution with mean  and standard deviation (  /n 0.5 ) as n is increased. Note: n, the number of replicates per sample, should be at least thirty.

Calculating a Confidence Interval Assume: n  30,  known

Effect of 1-   /2 1-   /2

Understanding What is a 95% Confidence Interval If we compute values of the confidence interval with many different random samples from the same population, then in about 95% of those samples, the value of the 95% c.i. so calculated would include the value of the population mean, . Note that  is a constant. The c.i. vary because they are each based on a sample.

Improving the Estimate of the Mean Reduce the confidence interval. Variables to examine: 1-  n 

Effect of 1-   /2 1-   /2

Effect of n (the sampling distribution of xbar)

Effect of 