Survey zHow would you judge the pace of the lectures? zDo you find the notes meaningful? zCan you offer any suggestions for improving the slide/lectures? zHave you found the labs meaningful for the intended objectives (e.g., graphical presentation and probability)? zCan you offer any suggestions for improvement for the labs?

Survey zIf you could have me go over one topic again what would it be? zIn the integers from 1 to 10, what is your favorite number?

Statistics Lecture 7

zDistribution of scores on a standardized test can be approximated by a normal distribution with mean of 500 and standard deviation of 100. Find probability that a randomly selected student scores: zOver 650 zBetween 325 and 675 zWhat proportion of students score better than 680?

Checking Normality zDoes normal distribution reasonably approximate distribution of data zCan use a normal probability plot (or normal scores plot) to assess normality zPlots sorted data versus percentiles of standard normal distribution zIf data is normally distributed, plot should display:

Example zIt is felt that the distribution of scores on a standardized test can be approximated by a normal distribution zTo see if this is true, a random sample of 15 students’ scores is taken

Sampling Distributions zA parameter is a numerical feature of a distribution or population zStatistic is a function of sample data

zSuppose you draw a sample and compute the value of a statistic zSuppose you draw another sample of the same size and compute the value of the statistic zWould the 2 statistics be equal?

zUse statistics to estimate parameters zWill the statistics be exactly equal to the parameter? zObserved value of the statistics depends on the sample zThere will be variability in the values of the statistic over repeated sampling

zProbability distribution of a statistic is called the sampling distribution (or distribution of the statistic) zBased on repeated random samples of the same size from the population zIn a random sample, the observations are independent and identically distributed

Example zLarge population is described by the probability distribution zIf a sample of size 2 is computed, what is the sampling distribution for the sample mean?

Sampling Distribution of the Sample Mean zHave a random sample of size n zThe sample mean is zWhat is it estimating?

Properties of the Sample Mean zExpected value: zVariance: zStandard Deviation:

Sampling from a Normal Distribution zSuppose have a sample of size n from a distribution zWhat is distribution of the sample mean?

Example zDistribution of moisture content per pound of a dehydrated protein concentrate is normally distributed with mean 3.5 and standard deviation of 0.6. zRandom sample of 36 specimens of this concentrate is taken zDistribution of sample mean? zWhat is probability that the sample mean is less than 3.5?

Central Limit Theorem zIn a random sample (iid sample) from any population with mean and standard deviation when n is large, the distribution of the sample mean is approximately normal. zThat is, zThus,

Implications zSo, for random samples, if have enough data, sample mean is approximately normally distributed...even if data not normally distributed zIf have enough data, can use the normal distribution to make probability statements about

Example zA busy intersection has an average of 2.2 accidents per week with a standard deviation of 1.4 accidents zSuppose you monitor this intersection of a given year, recording the number of accidents per week. zData takes on integers (0,1,2,...) thus distribution of number of accidents not normal. zWhat is the distribution of the mean number of accidents per week based on a sample of 52 weeks of data

Example zWhat is the approximate probability that is less than 2 zWhat is the approximate probability that there are less than 100 accidents in a given year?