Using the Central Limit Theorem Quiz

PDF page 318: Law of Large Numbers The law of large numbers says that if you take samples of larger and larger size from any population, then the mean of the sampling distribution, μ – tends to get closer and closer to the true population mean, μ. From the Central Limit Theorem, x we know that as n gets larger and larger, the sample means follow a normal distribution. The larger n gets, the smaller the standard deviation of the sampling distribution gets. (Remember that the standard deviation for the sampling distribution – –

PDF page 318: Law of Large Numbers The law of large numbers says that if you take samples of larger and larger size from any population, then the mean of the sampling distribution, μ – tends to get closer and closer to the true population mean, μ. From the Central Limit Theorem, x we know that as n gets larger and larger, the sample means follow a normal distribution. The larger n gets, the smaller the standard deviation of the sampling distribution gets. (Remember that the standard deviation for the sampling distribution – –

PDF page 318: Law of Large Numbers The law of large numbers says that if you take samples of larger and larger size from any population, then the mean of the sampling distribution, μ – tends to get closer and closer to the true population mean, μ. From the Central Limit Theorem, x we know that as n gets larger and larger, the sample means follow a normal distribution. The larger n gets, the smaller the standard deviation of the sampling distribution gets. (Remember that the standard deviation for the sampling distribution – –

PDF page 324: sample means for each distribution. As the sample size increases, n goes from 10 to 30 to 50, the standard deviations of the respective sampling distributions decrease because the sample size is in the denominator of the standard deviations of the sampling distributions.

PDF page 318: σ of X is .) This means that the sample mean x must be closer to the population mean μ as n increases. We can say n that μ is the value that the sample means approach as n gets larger. The Central Limit Theorem illustrates the law of large numbers. This concept is so important and plays such a critical role in what follows it deserves to be developed further. Indeed, there are two critical issues that flow from the Central Limit Theorem and the application of the Law of Large numbers to it. These are

PDF page 318: σ of X is .) This means that the sample mean x must be closer to the population mean μ as n increases. We can say n that μ is the value that the sample means approach as n gets larger. The Central Limit Theorem illustrates the law of large numbers. This concept is so important and plays such a critical role in what follows it deserves to be developed further. Indeed, there are two critical issues that flow from the Central Limit Theorem and the application of the Law of Large numbers to it. These are

PDF page 318: three panels show the histograms for 1,000 randomly drawn samples for different sample sizes: n=10, n= 25 and n=50. As

PDF page 318: three panels show the histograms for 1,000 randomly drawn samples for different sample sizes: n=10, n= 25 and n=50. As

PDF page 318: the sample size increases, and the number of samples taken remains constant, the distribution of the 1,000 sample means becomes closer to the smooth line that represents the normal distribution.

PDF page 318: Figure 7.3 is for a normal distribution of individual observations and we would expect the sampling distribution to

PDF page 318: converge on the normal quickly. The results show this and show that even at a very small sample size the distribution is close to the normal distribution.

PDF page 322: of success creating the skew in the distribution. For skewed distributions our intuition would say that this will take larger sample sizes to move to a normal distribution and indeed that is what we observe from the simulation. Nevertheless, at a sample size of 50, not considered a very large sample, the distribution of sample means has very decidedly gained the shape of the normal distribution.

PDF page 322: of success creating the skew in the distribution. For skewed distributions our intuition would say that this will take larger sample sizes to move to a normal distribution and indeed that is what we observe from the simulation. Nevertheless, at a sample size of 50, not considered a very large sample, the distribution of sample means has very decidedly gained the shape of the normal distribution.

PDF page 324: The Central Limit Theorem provides more than the proof that the sampling distribution of means is normally distributed. It also provides us with the mean and standard deviation of this distribution. Further, as discussed above, the expected value of the mean, μ – , is equal to the mean of the population of the original data which is what we are interested in estimating x from the sample we took. We have already inserted this conclusion of the Central Limit Theorem into the formula we use for standardizing from the sampling distribution to the standard normal distribution. And finally, the Central Limit Theorem σ