Section 2.2: Normal Distributions Abby Knowles, Irina Erickson, Ava Ciriello

Important Vocabulary Normal Curves: An important class of density curves that are symmetric, single-peaked, and bell-shaped Normal Distribution: displayed by a normal density curve, completely specified by two numbers, (1) its mean μ and (2) its standard deviation σ Standard Normal Distribution: the normal distribution with mean 0 and standard deviation 1 The Standard Normal Table (table A): a table of areas under the standard normal curve; the number in the table for each value of z is the area under the curve to the left of z Normal Probability Plot: used to assess whether a data set follows a normal distribution

Important Concept: Characteristics of Normal Curves ALL normal curves have the same shape Symmetric, Single-peaked, bell-shaped μ: mean of population σX: standard deviation of population A normal distribution is described by a normal density curve the mean is located in the center and is the same as the median

Describe chance outcomes (i.e. coin toss results) Important Concept: Why is a normal distribution important in statistics? Normal distributions occur in real life situations such as test scores, measurements of the same quantity, and characteristics of biological populations Describe chance outcomes (i.e. coin toss results) Statistical inferences can be made based on normal distributions

Important Concept: The 68-95-99.7 Rule The Empirical Rule, or 68-95-99.7 Rule, states that in a Normal distribution with mean μ and standard deviation σ: Approximately 68% of the observations fall within σ (1 standard deviation) of the mean μ. Approximately 95% of the observations fall within 2σ (2 standard deviations) of μ. Approximately 99.7% of the observations fall within 3σ (3 standard deviations) of μ.

The 68-95-99.7 Rule

Important Concept: Z-Scores How to Calculate: z = (x-μ)/σ **z = 1 corresponds to one standard deviation above the mean on a standard normal curve, and z = -1 corresponds to one standard deviation below the mean on a standard normal curve**

How to Use a Z-score Once you have calculated the z-score of a specific observation using the mean and standard deviation , you can look up the z-score on the standard normal table (found on our blue sheets). The entry on the table that corresponds with the z-score is the area underneath the standard normal curve to the left of z This area is the proportion of observations that are less than z To find the proportion of observations that are greater than (to the right of) z, you can: look up the z-score as it is, then subtract the corresponding table value from 1 (because the total area under the curve = 1) look up the inverse of the z-score, then use the corresponding table value as it is

Z-Scores

Important Concept: Normal Probability Plot A normal probability plot shows if the data follows a normal distribution (Y axis: z-score; X axis: observed data values) Straight line: Data follows normal distribution Not a Straight line: Data is not normally distributed

How to Make a Normal Probability Plot Arrange the data values from smallest to largest and record the percentile of each observation Use the standard normal distribution to find the z-scores at these percentiles Plot each observation x against the corresponding z

Normal Probability Plots Normal Distribution not Normally Distributed ZScore ZScore Data Data

Applying Concepts The figure below is a Normal probability plot of the heart rates of 200 male runners after six minutes of exercise on a treadmill. The distribution is close to Normal. How can you see this? Describe the nature of the small deviations from Normality that are visible in the plot.

Correct Answer The plot is nearly linear. The smallest value is smaller than we would expect and the largest two values are larger than we would expect. There is also a cluster of points around 125 bpm that are a little larger than expected.

Applying Concepts The adhesion of one 4400-horsepower diesel locomotive varies in actual use according to a Normal distribution with mean μ = 0.37 and standard deviation σ = 0.04.For each part that follows, sketch and shade an appropriate Normal distribution. (a) For a certain small train’s daily route, the locomotive needs to have an adhesion of at least 0.30 for the train to arrive at its destination on time. On what proportion of days will this happen? (b) An adhesion greater than 0.50 for the locomotive will result in a problem because the train will arrive too early at a switch point along the route. On what proportion of days will this happen?

Correct Answers (a) We would expect trains to arrive on time about 96% of the time. (b) z = 3.25. We would expect trains to arrive early 0.06% of the time.