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Survey How would you judge the pace of the lectures? Do you find the notes meaningful? Can you offer any suggestions for improving the slide/lectures? Have you found the labs meaningful for the intended objectives (e.g., graphical presentation and probability)? Can you offer any suggestions for improvement for the labs? Survey If you could have me go over one topic again what would it be? In the integers from 1 to 10, what is your favorite number? Statistics 400 - Lecture 7 Distribution 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: Over 650 Between 325 and 675 What proportion of students score better than 680? Checking Normality Does normal distribution reasonably approximate distribution of data Can use a normal probability plot (or normal scores plot) to assess normality Plots sorted data versus percentiles of standard normal distribution If data is normally distributed, plot should display: Example It is felt that the distribution of scores on a standardized test can be approximated by a normal distribution To see if this is true, a random sample of 15 students’ scores is taken 403 633 315 630 505 711 546 561 574 420 428 474 473 489 565 P T e s t S c o r 30 40 50 60 70 No r ma l -1 0 1 Qu a n t i l e s o Sampling Distributions A parameter is a numerical feature of a distribution or population Statistic is a function of sample data Suppose you draw a sample and compute the value of a statistic Suppose you draw another sample of the same size and compute the value of the statistic Would the 2 statistics be equal? Use statistics to estimate parameters Will the statistics be exactly equal to the parameter? Observed value of the statistics depends on the sample There will be variability in the values of the statistic over repeated sampling Probability distribution of a statistic is called the sampling distribution (or distribution of the statistic) Based on repeated random samples of the same size from the population In a random sample, the observations are independent and identically distributed Example Large population is described by the probability distribution X P(X=x) 0 0.2 3 0.3 12 0.5 If a sample of size 2 is computed, what is the sampling distribution for the sample mean? Sampling Distribution of the Sample Mean Have a random sample of size n The sample mean is n xi x i1 n What is it estimating? Properties of the Sample Mean Expected value: Variance: Standard Deviation: Sampling from a Normal Distribution Suppose have a sample of size n from a N ( , ) distribution What is distribution of the sample mean? Example Distribution of moisture content per pound of a dehydrated protein concentrate is normally distributed with mean 3.5 and standard deviation of 0.6. Random sample of 36 specimens of this concentrate is taken Distribution of sample mean? What is probability that the sample mean is less than 3.5? Central Limit Theorem In 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. x That is, Thus, x Z / n Implications So, for random samples, if have enough data, sample mean is approximately normally distributed...even if data not normally distributed If have enough data, can use the normal distribution to make probability statements about x Example A busy intersection has an average of 2.2 accidents per week with a standard deviation of 1.4 accidents Suppose you monitor this intersection of a given year, recording the number of accidents per week. Data takes on integers (0,1,2,...) thus distribution of number of accidents not normal. What is the distribution of the mean number of accidents per week based on a sample of 52 weeks of data Example What is the approximate probability that x is less than 2 What is the approximate probability that there are less than 100 accidents in a given year?
