Which Class Did Better? Group 12A RESULTS(%) 0< x  2020< x  3030< x  5050< x  7070< x  100 FREQUENCY Group 12B RESULTS(%) 0< x  2020< x  3030< x  5050< x  7070< x  100 FREQUENCY Finding the variance and standard deviation of a set of numbers(recap)

Estimate 1 Introduction to sampling Unbiased estimators of the population mean and variance. Sampling Standard Error.

How could we estimate the mean and standard deviation of all the students at Bilborough College? For example: If we found the mean and standard deviation of the heights of students in this class then we could use those results to estimate the mean and standard deviation of the heights of all students in Bilborough college. sx to σx sx x √(n-1/n) = σx We could use sampling.

Samples and populations The sample mean x is an unbiased estimate of the population mean μ. The sample variance s 2 is an unbiased estimate of the population variance σ 2. The random variable X is an unbiased estimate of the population mean μ. The random variable S 2 is an unbiased estimate of the population variance σ 2.

Standard error The standard error of the mean is given by σ/√n, which is estimated by s/√n if σ is not known Mini Whiteboards Quick questions What do you think happens if the sample size is increased?

0.712

0.733

0.616

0.625

Standard error The standard error of the mean is given by σ/√n, which is estimated by s/√n if σ is not known Now do Ex 5A page 122 What do you think happens if the sample size is increased? Sometimes you may be asked to estimate the standard error of a population based on a sample. To do this you must use sx instead σ x. So the estimated variance for the population would be sx 2.

6 sided die Throw 2 times and find mean Repeat as many times as you can in 2 mins and record for all mean.

6 sided die Throw 8 times and find mean. Repeat as many times as you can in 2 mins and record for all mean.

Introduction to the sample mean distribution A population has the following values: 1, 2, 3, 4, 5, 6 How could we take random samples of size 2 from this population? If we find the mean of all such samples we generate another distribution called the distribution of the sample mean

Population samples are taken with sample size 5 Mean = 16 s.d = 9.52 Parent population Mean = s.d = 4.20

The population is a normal distribution with a mean of 16 and standard deviation of samples are taken of sample size 25 Mean = 16 s.d = 5 Mean = s.d = 0.99 Parent population

The distribution of the sample mean will be a normal distribution For a population that has a normal distribution with a mean equal to the mean of the population and a variance equal to the variance of the population divided by the sample size In symbols this reads: If X ~ N( μ, σ 2 ) The distribution of the sample mean is given by: X ~ N( μ, σ 2 /n )

Ex 5B page 125 q1 A random variable has a normal distribution with a mean of 12 and a standard deviation of 3. A random sample of 81 observations are taken. (a) Find the mean and variance of the distribution of the sample mean. (b) Find the probability that the sample mean is less than 11.5

The weight of a soldier may be taken to be an observation from a normal distribution with a mean of 90kg and standard deviation 10kg. There are 250 soldiers on board. (a) Find the mean and variance of the average weight of the soldiers on the aircraft. State any assumption necessary. (b) Find the probability that the mean weight lies between 89kg and 91kg. Ex 5B page 125 q2

A random variable X has a normal distribution with a mean of 150 and a standard deviation of 2. A random sample of n observations of X is taken. Find the smallest value of n such that the standard deviation of the sample mean is less than 0.1. Ex 5B page 125 q4

Mean and variance It can be shown that if a random sample consists of n observations of a random variable X, and the mean X is found, then E(X)=µ and Var(X)=σ 2 n