One of the most basic types of problems in statistics involves the use of the normal distribution, or bell curve. Here if we know that we are working with a normal distribution we can use a table of z scores to answer questions concerning the proportion. In particular we are interested in finding the proportion of the population greater than or less than a particular observed value.
We will work through several types of these problems to see the thought process that is required.
Given that the heights of a population of men are normally distributed with mean of 68 inches and standard deviation of 2 inches, what proportion of men are less than 65 inches tall?
We begin by stating what we know. We have a mean of 68, a standard deviation of 2 and an observed value of 65. The proportion of men with height less than 65 inches is unknown. In other words, the area under the particular normal distribution to the left of 65 is what is unknown.
We standardize the problem by converting our observed value to a z-score. We see that z = (65 – 68)/2 = -1.5. This means that the observed value is one and a half standard deviations below the mean. A table of z scores or the use of statistical software, such as NORM.DIST in Excel shows that approximately 6.68% of men are less than 65 inches tall.
Given that the heights of a population of men are normally distributed with mean of 68 inches and standard deviation of 2 inches, how tall does a man need to be to be in the tallest 1% of men in the population?
We again start begin by stating what we know. We have a mean of 68, a standard deviation of 2 and an area of 1% above or 99% below an observed value. This observed value is unknown, and is what we need to determine.
We first find that the z score corresponding to the point of the distribution with 99% below that point. Software shows that the value of z = 2.326. We then use the equation for a z score to find the observed value 2.326 = (x - 68)/2. Therefore x = 72.65 inches.
Given that the heights of a different population of men are normally distributed with standard deviation of 2 inches, and we know that someone who is 69 inches is taller than 80% of all men in this population, what is the mean height of the population?
We again start begin by stating what we know. We have a standard deviation of 2, and an area of 20% above or 80% below an observed value of 69. The mean in this problem is unknown, and is what we will figure out.
We use software or a table to show that the z score that corresponds with the point on the distribution that we described is z = .8416. We then use our z score formula to determine the unknown mean: .8416 = (69 – mean)/2. Therefore the mean is equal to 67.32 inches.
We have just seen the same problem from three different angles. In each case we knew a different set of values and there was a different unknown. By methodically writing down your thoughts, it can make solving problems such as these much more easy and enjoyable.