STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 5 6 mean Population standard deviation X1 X2 Xn Random Sample X1 X2 Xn or E X 1 X 2 X n n are i i d SD X 1 X 2 X n n If the sampling is done without replacement from a finite population of size N N n then SD X n N 1 X The sample mean X1 X 2 X n n E X SD X n If the sampling is done without replacement from a finite population of size N N n then SD X N 1 n X chance error LAW OF LARGE NUMBERS LAW As the sample size n AVERAGES increases the sample mean X tends to gets closer and closer to the population mean As the number of trials OF n increases the sample proportion of successes X tends to gets closer and closer to the probability of success p n CENTRAL LIMIT THEOREM If the sample size n is large the sampling distribution of the sample total is approximately normal with mean n and standard deviation n X n Z n Therefore If the population itself is normally distributed the sampling distribution of the sample total is normal for any sample size n If the sample size n is large the sampling distribution of the sample mean X is approximately normal with mean and standard deviation n X Therefore Z n If the population itself is normally distributed the sampling distribution of the sample mean X is normal for any sample size n Case 1 Any population n large Case 2 Normal population Any n Case 3 Population NOT Normal n small 1 A student commission wants to know the mean amount of money spent by college students for textbooks in one semester Suppose the population mean is 450 and the population standard deviation is 40 A random sample of 625 students is taken a What is the probability that the sample mean will be less than 452 b What is the probability that the sample mean will be within 2 of 450 That is what is the probability that the sample mean will be between 448 and 452 c What is the probability that the sample mean will be within 10 of 450 That is what is the probability that the sample mean will be between 440 and 460 2 The amount of sulfur in the daily emissions from a power plant has a normal distribution with mean of 134 pounds and a standard deviation of 22 pounds For a random sample of 5 days find the probability that the total amount of sulfur emissions will exceed 700 pounds 3 An economist wishes to estimate the average family income in a certain population The population standard deviation is known to be 4 500 and the economist uses a random sample of 225 families What is the probability that the sample mean will fall within 600 of the population mean 4 Forty eight measurements are recorded to several decimal places Each of these 48 numbers is rounded off to the nearest integer The sum of the original 48 numbers is approximated by the sum of these integers If we assume that the errors made by rounding off are i i d and have uniform distribution over the interval 12 12 compute approximately the probability that the sum of the integers is within 2 units of the true sum
View Full Document
Unlocking...