A confidence interval for a population mean, when the population standard deviation is known based on the conclusion of the Central Limit Theorem that the sampling distribution of the sample means follow an approximately normal distribution. Calculating the Confidence Interval…
Figure 8.1 Have you ever wondered what the average number of M&Ms in a bag at the grocery store is? You can use confidence intervals to answer this question. (credit: modification of work “sweet, orange, food, green, red, color, brown, blue, colorful,…
The Central Limit Theorem for Sample Means In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the…
We saw that the sample size has an important effect on the variance and thus the standard deviation of the sampling distribution. Also of interest is the proportion of the total population that has been sampled. We have assumed that…
The Central Limit Theorem tells us that the point estimate for the sample mean, x¯x¯, comes from a normal distribution of x¯x¯’s. This theoretical distribution is called the sampling distribution of x¯x¯’s. We now investigate the sampling distribution for another important parameter we…
The Law of Large Numbers, along with the Central Limit Theorem, provides another critical piece of information to allow us to engage in inferential statistics. In short, the Law of Large Numbers proves that the expected value of the sampling distribution…
The sampling distribution is a theoretical distribution. It is created by taking many samples of size n from a population. Each sample mean is then treated like a single observation of this new distribution, the sampling distribution. The genius of…
Figure 7.1 If you want to figure out the distribution of the change people carry in their pockets, using the Central Limit Theorem and assuming your sample is large enough, you will find that the distribution is the normal probability density function.…
The Standard Normal Distribution A z-score is a standardized value. Its distribution is the standard normal, Z ~ N(0, 1). The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution N(µ, σ) then z tells you how many…
We found earlier that various probability density functions are the limiting distributions of others; thus, we can estimate one with another under certain circumstances. We will find here that the normal distribution can be used to estimate a binomial process.…