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.

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Presentation transcript:

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.

Standard error The standard error of the mean is given by σ/√n, which is estimated by s/√n if σ is not known 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.

Estimate 2 Introduction to the sample mean distribution Investigating the sample mean distribution for a normal population Practice questions The Central Limit theorem Exam questions Central Limit Theorem

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

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

The distribution of the sample mean will be a normal distribution Recap: 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 )

Central Limit Theorem If X ~any distribution with a mean of μ and a variance of σ 2 X ~ N( μ, σ 2 /n ) approximately The accuracy of the approximation depends on the shape of the original population and the sample size. We usually say n has to be bigger or equal to 30 for a good approximation.

Example 1 A random variable X has a mean 40 and variance 800. A random sample of 50 observations of X is taken and the mean, of each sample is calculated. How, approximately is X distributed?

Page 140 q5 A firm has cars available for hire. The distances travelled per day by a hire car has a mean of 236 miles and a standard deviation of 80 miles. (a) For a random sample of 100 such distances, determine the probability that its mean is less than 250 miles. (b) Name the theorem you have used and explain why it is applicable in this case. a)0.960 b)Central Limit Theorem, large sample (>30)

Tarsia puzzle in groups of 3-4