# Sampling Error and the Sampling Distribution of the Sample Mean

### We can then develop a histogram of the sample means!  This is what is called the ‘sampling distribution of the sample mean’.  We would expect the distribution of sample means to be less dispersed than the distribution of the ages of ALL incoming MEM students.  If the sample size is large enough, we would expect than the mean of the sample means would approach the true population mean.  In addition, as the sample size increases, the distribution of the means will approach the normal distribution.  With a large sample size, the sample means are normally distributed with a mean of μ (mu) and a standard deviation of σ/sqrt(n). The standard deviation of the sample means is called the standard error of the mean (σ/sqrt(n)).  In statistical notation: # The Central Limit Theorem!

### The essence of the Central Limit Theorem:  As the sample size increases, the sampling distribution of the sample mean ( xbar ) concentrates more and more around µ (the population mean).  The shape of the distribution also gets closer and closer to the normal distribution as sample size n increases.  An example follows.  The figure below shows a histogram of our (make believe) population. ### Typically the exact distribution of a population is UNKNOWN, but to demonstrate the Central Limit Theorem, we will start with this known distribution (in blue).  This population has a mean (mu) of 2.25 and a standard deviation of 3.93.  From this population distribution, I randomly selected a sample of 2 (n=2) and calculated an average (xbar).  I then repeated this for a total of 1,000 times and made a histogram of the 1,000 sample means. ### As you can see in the red histogram (sample size n=2), the dispersion of the distribution of sample means is less than the parent population (a greater concentration of values around the mean).  The empirical mean of this distribution is 2.31 with a standard deviation of 2.79.  I repeated this sampling process three more times with sample sizes of 5, 20 and 100 (see the histograms below).  As you can see, as sample size increases, the distribution gets increasingly narrow and increasingly approaches a normal distribution. This is the essence of the Central Limit Theorem.   # Calculating Z-Scores with the Sampling Distribution of the Sample Means ## Solution

### (2)  Because we are interested in just one bulb (not an average), we use this z-score formula: ### (3)  Because we are interested in the average, we use the following z-score formula: 