As you might have noticed, hipsters often wear oversized glasses, yet many hipsters don't really need glasses. The glasses only serve as a celebration as their hipsterness. Suppose you studied the strengths of the glasses of the hipsters in Italy. You know that the variable strength of glasses has a population distribution that is skewed to the left. The peak will be around 0 because most glasses are fake glasses. But, of course, not all glasses are fake because hipsters are generally relatively young and young people are more often short sighted than long sighted, there will be more hipsters who are short sighted than hipsters who are long sighted. Those who are short sighted have a negative score on strengths of glasses. And those who are long sighted have a positive score. You know that the mean in the population is -0.75 and that the standard deviation is 2.89. We would like to know three things. First, what does the population distribution look like? We would like to see shape, mean and standard deviation. Second, what does the sampling distribution of the sample mean look like based on a sample size of n equals 3,000? Third, what does the sample or data distribution look like if you draw a simple random sample of 3,000 cases? And fourth, what is the probability of selecting such a sample from this population with a sample mean between -0.71 and -0.81? Let's start with the first question. The distribution would look something like this. The peak is around 0, and the distribution is skewed to the left. The population mean is left of population mode. We know that the score of this parameter, symbolized by mu, is -0.75. The population standard deviation, symbolized by sigma, is 2.89. The second question is what the sampling distribution of the sample mean looks like. We know that when the sample size is sufficiently large which is with an n of 3,000, clearly the case in this example, the sampling distribution is bell-shaped with a mean that equals the population. We can therefore conclude that mu x-bar equals mu equals -0.75. The standard deviation of the sampling distribution symbolized by sigma x-bar can easily be computed. It is a standard deviation in the population divided by the square root of n, that is 2.89 divided by the square root of 3,000 equals 0.05. The cases in the sampling distribution of the sample mean are not individual cases as in the population distribution, but an infinite number of sample means. This is symbolized by the x-bar next to the mu and the sigma. The third question is what the sample or data distribution looks like. This is what the distribution of scores looks like in our sample of 3,000 Italian hipsters. Because we are dealing with a simple random sample with a fairly large sample size, we can be pretty confident that the sample resembles the population. The shape of its distribution will be very similar to the shape of the population distribution and the sample mean, symbolized by x-bar, will be close to the population mean of -0.75. The sample standard deviation symbolized by s would be closer to the population standard deviation of 2.89. However, also note that it is very unlikely that the sample statistics are exactly the same as the population parameters. You will probably have a sample mean of, say, -0.70, or -0.78. Similarly, your sample standard deviation will probably have a value slightly different from, but very close to 2.89. It could for instance be 2.77 or 3.01. In fact, based on the sampling distribution of the sample mean, we can compute what the probability is of finding particular sample means. The fourth question we want to answer is what the probability is of selecting a sample from this population with a sample mean between -0.71 and -0.81. We are dealing with a sample mean here so we should look at the sampling distribution. This is what the sampling distribution looks like. We're interested in the probability of finding a value between these two values, so we're interested in this surface. To find this probability we first have to convert the original scores of -0.71 and -0.81 into z scores. This is the relevant formula. Let's first look at the score of -0.71. We subtract -0.75 which is the mean of the sampling distribution from -0.71 which is the sample mean score we're interested in, and divide the outcome by the standard deviation of the sampling distribution. We have already computed this value, it's the population mean divided by the square root of the sample size. That equals 0.05. The outcome is 0.8. This means that the original score of -0.71 corresponds to a z score of 0.8. We also do that with the score of -0.81. So we subtract -0.75 from -0.81 and divide it by 0.05. That makes -1.2. So, the original scores of -0.81 corresponds to a z score of -1.2. We thus have to find the probability of finding a value between a z value of -1.2 and a z score of 0.8. If we look at the z table we see that the probability of finding a value lower than a z score of 0.8 is 0.7881. That corresponds to this surface. The probability of finding a value lower than the z score of 1.2 is 0.1151. That corresponds to this surface. We're interested in this surface, so we have to subtract this part from this part. That means 0.7881 minus 0.1151. That is about 0.67, or 67%. Conclusion, the probability of selecting a sample of n equals 3,000 with a sample mean between -0.71 and -0.81 is 67%.