Read Online Statistics Essentials For Dummies by Deborah Rumsey - Free Book Online Page A
Authors: Deborah Rumsey
Tags: Reference
Ads: Link
Then do the same with Pond #2. Knowing that the fish in Pond #2 have more variability than Pond #1 in the first place, the means of the samples from Pond #2 will have more variability compared to Pond #1 as well. It's harder to estimate the population average when the population varies a lot to begin with — it's much easier to estimate the population average when the population values are similar.
Figure 6-3: Distributions of a) fish lengths in Pond #1; b) in Pond #2.
The shape
Now that we know the mean and standard error of , the next step is to determine the sampling distribution of (that is, the shape of the distribution of all possible 's from all possible samples). There are two cases: 1) the original distribution for X (the population) is normal; and 2) the original distribution for X (the population) is not normal, or is unknown.
Case 1: Distribution of X is normal
If X has a normal distribution, then does too. This is a mathematical statistics result and requires no additional tools to prove. Looking at Figure 6-2, you can see this result is true for the worker's times. Since X is normal, the shape is the same in each graph; the only thing that changes is the amount concentration around the mean.
Case 2: Distribution of X is unknown or not normal
If the X distribution is any distribution that is not normal, or if its distribution is unknown, you can't automatically say the sample means ( ) have a normal distribution. But you can approximate 's distribution with a normal distribution — if the sample size is large enough. This result is due to the Central Limit Theorem (CLT). The CLT says that the sampling distribution (shape) of is approximately normal, if the sample size is large enough. And the CLTdoesn't care what the distribution of X is!
Formally, for any population with mean and standard deviation , the CLT states that:
If the distribution of is non-normal or unknown, the sampling distribution of all possible sample means, is approximately normal for a sufficiently large sample size.
The larger the sample size ( n ), the closer the distribution of the sample means will be to a normal distribution.
Most statisticians agree that if n is at least 30, it will do a reasonable job in most cases.
Two common misconceptions about the CLT:
The CLT is only needed when the distribution of X is either non-normal or is unknown. It is not needed if X started out with a normal distribution.
The formulas for the mean and standard error of are not due to the CLT. These are just mathematical results that are always true.
Finding Probabilities for
After you've established through Case 1 or Case 2 (see previous section) that has a normal or approximately normal distri-bution, you can find probabilities for by converting the -value to a z -value and finding probabilities using the Z -table (Table A-1
in the appendix.) The general conversion formula is .
Substituting the appropriate values of the mean and standard
error of the conversion formula becomes .
Suppose X is the time it takes a worker to type and send 5 letters of recommendation. Suppose X (the times for all the workers) has a normal distribution and the reported mean is 10 minutes and the standard deviation 2 minutes. You take a random sample of 50 workers and measure their times. What is the chance that their average time is less than 9.5 minutes?
This question translates to finding P( < 9.5). As X has a normal distribution to start with, we know also has a normal
distribution. Converting to z -value. we get .
So we want P( Z < -1.77), which equals 0.0384 from the Z-table (Table A-1 in the appendix). So the chance that these 50 randomly selected workers average less than 9.5 minutes to complete this task is 3.84%.
Don't forget to divide by the square root of n in the denominator of Z . Always divide by square root of n when the question refers to the average of the X- values.
How do you find probabilities for if X is not normal, or is unknown?
Figure 6-3: Distributions of a) fish lengths in Pond #1; b) in Pond #2.
The shape
Now that we know the mean and standard error of , the next step is to determine the sampling distribution of (that is, the shape of the distribution of all possible 's from all possible samples). There are two cases: 1) the original distribution for X (the population) is normal; and 2) the original distribution for X (the population) is not normal, or is unknown.
Case 1: Distribution of X is normal
If X has a normal distribution, then does too. This is a mathematical statistics result and requires no additional tools to prove. Looking at Figure 6-2, you can see this result is true for the worker's times. Since X is normal, the shape is the same in each graph; the only thing that changes is the amount concentration around the mean.
Case 2: Distribution of X is unknown or not normal
If the X distribution is any distribution that is not normal, or if its distribution is unknown, you can't automatically say the sample means ( ) have a normal distribution. But you can approximate 's distribution with a normal distribution — if the sample size is large enough. This result is due to the Central Limit Theorem (CLT). The CLT says that the sampling distribution (shape) of is approximately normal, if the sample size is large enough. And the CLTdoesn't care what the distribution of X is!
Formally, for any population with mean and standard deviation , the CLT states that:
If the distribution of is non-normal or unknown, the sampling distribution of all possible sample means, is approximately normal for a sufficiently large sample size.
The larger the sample size ( n ), the closer the distribution of the sample means will be to a normal distribution.
Most statisticians agree that if n is at least 30, it will do a reasonable job in most cases.
Two common misconceptions about the CLT:
The CLT is only needed when the distribution of X is either non-normal or is unknown. It is not needed if X started out with a normal distribution.
The formulas for the mean and standard error of are not due to the CLT. These are just mathematical results that are always true.
Finding Probabilities for
After you've established through Case 1 or Case 2 (see previous section) that has a normal or approximately normal distri-bution, you can find probabilities for by converting the -value to a z -value and finding probabilities using the Z -table (Table A-1
in the appendix.) The general conversion formula is .
Substituting the appropriate values of the mean and standard
error of the conversion formula becomes .
Suppose X is the time it takes a worker to type and send 5 letters of recommendation. Suppose X (the times for all the workers) has a normal distribution and the reported mean is 10 minutes and the standard deviation 2 minutes. You take a random sample of 50 workers and measure their times. What is the chance that their average time is less than 9.5 minutes?
This question translates to finding P( < 9.5). As X has a normal distribution to start with, we know also has a normal
distribution. Converting to z -value. we get .
So we want P( Z < -1.77), which equals 0.0384 from the Z-table (Table A-1 in the appendix). So the chance that these 50 randomly selected workers average less than 9.5 minutes to complete this task is 3.84%.
Don't forget to divide by the square root of n in the denominator of Z . Always divide by square root of n when the question refers to the average of the X- values.
How do you find probabilities for if X is not normal, or is unknown?
Similar Books
