CHAPTER 2 Modeling Distributions of Data 2.2 (part 1) Density Curves and Normal Distributions

Exploring Quantitative Data In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy. Exploring Quantitative Data Always plot your data: make a graph, usually a dotplot, stemplot, or histogram. Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

Normal Distributions One particularly important class of density curves are the Normal curves, which describe Normal distributions. All Normal curves have the same shape: symmetric, single-peaked, and bell-shaped Any specific Normal curve is completely described by giving its mean µ and its standard deviation σ.

Normal Distributions A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. The mean of a Normal distribution is the center of the symmetric Normal curve. The standard deviation is the distance from the center to the change-of-curvature points on either side. We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

The 68-95-99.7 Rule Although there are many Normal curves, they all have properties in common. The 68-95-99.7 Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

Examples: The distribution of heights of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. Draw a normal curve on which this mean and standard deviation are correctly located.

Use the “68-95-99.7% Rule” to answer the following questions: What percent of men are taller than 74 inches? Between what heights do the middle 95% of men fall? What percent of men are shorter than 66.5 inches? A height of 71.5 inches corresponds to what percentile of adult male American heights?

1. Items produced by a manufacturing process are supposed to weight 90 grams. The manufacturing process is such, however, that there is variability in the items produced and they do not all weigh exactly 90 grams. The distribution of weights can be approximated by a normal distribution with mean 90 grams and a standard deviation of 1 gram. Using the 68-95-99.7 rule, what percentage of the will either weigh less than 87 grams or more than 93 grams?

2. The time to complete a standardized exam is approximately normal with a mean of 70 minutes and a standard deviation of 10 minutes. Using the 68-95-99.7 rule, what percentage of students will complete the exam in under an hour?

Empirical Rule predicts about mpg. 3. Guzzlers? Environmental Protection agency (EPA) fuel economy estimates for automobile models tested recently predicted a mean of 24.8 mpg and a standard deviation of 6.2 mpg for highway driving. Assume the Normal model can be applied. a) Draw the model for auto fuel economy. Clearly label it showing what the Empirical Rule predicts about mpg. b) In what interval would you expect the central 68% of all autos to be found? c) About what percent of autos should get more than 31 mpg? d) About what percent of cars should get between 31 and 37 mpg? e) Describe the gas mileage of the worst 2.5% of all cars.