To find probabilities in between two standard deviations, we must put them in terms of the probabilities below. (d) What is the probability of a normal random variable taking a value between 1 standard deviation below and 1 standard deviation above its mean? (a) What is the probability of a normal random variable taking a value less than 2.8 standard deviations above its mean? The columns fine-tune these values to hundredths, allowing us to look up the probability of being below any standardized value z of the form *.**.įor example, in the part of the table shown below, we can see that for a z-score of -2.81, we would find P(Z < -2.81) = 0.0025.īy construction, the probability P(Z < z*) equals the area under the z curve to the left of that particular value z*. These particular values are listed in the form *.* in rows along the left margins of the table, specifying the ones and tenths. The normal table provides probabilities that a standardized normal random variable Z would take a value less than or equal to a particular value z*. The normal table outlines the precise behavior of the standard normal random variable Z, the number of standard deviations a normal value x is below or above its mean. Also, according to the Standard Deviation Rule, most of the area under the standardized curve falls between z = -3 and z = +3. Since the total area under any normal curve is 1, it follows that the areas on either side of z = 0 are both 0.5. Since normal curves are symmetric about their mean, it follows that the curve of z scores must be symmetric about 0. Part (c) above illustrates how z-scores become crucial when you want to compare distributions.Note that even though Ross’ foot is longer than Candace’s, Candace’s foot is longer relative to their respective genders. Ross: z-score = (13.25 – 11) / 1.5 = 1.5 (Ross’ foot length is 1.5 standard deviations above the mean foot length for men).Ĭandace: z-score = (11.6 – 9.5) / 1.2 = 1.75 (Candace’s foot length is 1.75 standard deviations above the mean foot length for women). To answer this question, let’s find the z-score of each of these two normal values, bearing in mind that each of the values comes from a different normal distribution. Which of the two has a longer foot relative to his or her gender group? Ross’ foot length is 13.25 inches, and Candace’s foot length is only 11.6 inches. Assume that women’s foot length follows a normal distribution with a mean of 9.5 inches and standard deviation of 1.2. (c) In general, women’s foot length is shorter than men’s. Note that z-scores also allow us to compare values of different normal random variables. Since the mean is 11, and each standard deviation is 1.5, we get that the man’s foot length is: 11 + 2.5(1.5) = 14.75 inches. If z = +2.5, then his foot length is 2.5 standard deviations above the mean. What is his actual foot length in inches? (b) A man’s standardized foot length is +2.5. This foot length is 1.67 standard deviations below the mean. (a) What is the standardized value for a male foot length of 8.5 inches? How does this foot length relate to the mean? Details for Non-Parametric Alternatives in Case C-Q.Unit 4A: Introduction to Statistical Inference.
Summary (Unit 3B – Sampling Distributions).Sampling Distribution of the Sample Mean, x-bar.Sampling Distribution of the Sample Proportion, p-hat.Conditional Probability and Independence.Linear Relationships – Linear Regression.Standard Normal Distribution » Biostatistics » College of Public Health and Health Professions » University of Florida