Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Central Limit Theorem • We know from our study of populations that if a population has a mean and a standard deviation then it can be normally distributed. • The central limit deals with what happens when we take a sample from this population. Central Limit Theorem • The mean of an extremely large sample is very hard to obtain. • Therefore the easiest thing to do is to divide the population into different samples and get the mean of these samples. • The mean of these smaller groups will then be approximately the same as getting the mean of the population. Population Population Large Sample Sample Means Mean 𝜇 𝑥 𝜇𝑥 Standard Deviation 𝜎 𝑠 𝜎𝑥 (Standard Error) Population S ample 1 S ample 2 S ample 3 S ample 6 S ample 4 S ample 5 Central Limit theorem • The central limit theorem tells us that if the sample we take is large (n > 30) and we get the means of these samples they will form a normal distribution. • And the mean of these samples will be the same as the mean of the original population. • In simple English. The mean of the samples form a normal distribution when the sample size is large Central Limit Theorem • However the standard deviation of the samples will not be the same as the standard deviation of the population • So we must account for this error. Notes on the central limit theorem • The means of all the samples are sometimes refereed to as the sampling distribution. • We can still use the central limit theorem for samples less than 30 as long it states that the population is normally distributed. Hom e The distribution of sample means • The centre of the distribution (𝝁𝒙 ) is identical to the population centre (𝝁). Hom e The distribution of sample means • The mean of the samples is the same as the mean of the population Hom e The distribution of sample means • The distribution is more compact than the population. Hom e The distribution of sample means Why is the distribution of sample means more compact than the population? Hom e The distribution of sample means How does the spread of the distribution of sample means compare to the population? 𝟐. 𝟓𝟕 ≅𝟖 𝟎. 𝟑𝟐 𝝈 𝝈𝒙 = 𝒏 Summary of the Central Limit Theorem • The mean of random samples can be approximated by a Normal Distribution. • The larger the sample, the better the approximation will be • Basically, when you split a population into lots of smaller groups of equal size, if you get the mean of all their means, it will equal the mean of the whole group. • However, the standard deviation of the means of these groups is different to the overall standard deviation. Summary Population Large Sample Sample Means Mean 𝜇 𝑥 𝜇𝑥 Standard Deviation 𝜎 𝑠 𝜎𝑥 (Standard Error) In practice, from the table above, we can say that for n 30 1. The sample means are normally distributed. 2. The mean of the sample means is the same as the population mean. x 3. The standard deviation of the sample means is equal to this is called the standard error. x n n Question A random sample of size 49 is chosen from a population mean that is known to be normal. The mean of the population is 9 with a standard deviation of 2. Find the probability that the sample mean is greater than 10.