Guide to Chapter 7

This is your guide to what’s important in the chapter, with comments on some things that the chapter leaves out or doesn’t explain well. Page numbers refer to Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011), which is equivalent to the “second custom edition” for TC3.

Always check Corrections to Sullivan’s Fundamentals of Statistics, 3rd Edition for known mistakes in the textbook. Caution: your calculator is more accurate than the book because the book rounds values and uses tables. When the book disagrees with your answer, check the errata.

Overview: After doing discrete PD last week, we turn to continuous PD, specifically to the normal distribution, the bell curve. We study the ND for two reasons. First, many things in real life follow the normal distribution (ND). Second, we’ll discover in Chapter 8 that through sampling we can use a ND even with variables that aren’t ND.

Handout: Instead of all the calculations and table lookups in your book, use section 1 of Normal Calculations on TI-83/84 or TI-89.

MATH200A: Use MATH200A part 4 to test for normality in sec 7.4.

7.1  Properties of the Normal Distribution

325 ★ Continuous distributions no longer have a probability for a specific outcome. Instead there’s a probability that the outcome is within a range of x values. The discrete probability histogram smoothes out and becomes a continuous density curve. The height of that curve is called a pdf (probability density function).

329 ★ Figure 8 shows a good example of passing from a histogram to a density curve. In sec 7.4 we’ll see how to determine whether the data can be properly modeled as a ND.

326 ★ The purple box is true of all continuous or discrete distributions: area represents probability

We met “area = probability” in Chapter 3 with the Empirical Rule. This week we extend that to any region in the normal curve, not just 1, 2, or 3 s.d. from the mean.

325–6 Work through Examples 1 and 2 to understand the general idea of probability for a continuous random variable. However, after those examples this course doesn’t do anything more with the uniform distribution. Our focus is the normal distribution.

327 ★ Unlike any other distribution that we study, the normal curve is completely specified by its mean and standard deviation. See Figure 5 for the role of μ and σ in the graph of the ND.

328 ★ Know the properties of the normal curve.

329 height is approximately normal — real-life distributions aren’t exactly normal because of sampling error (Chapter 1)

330 purple box: area = proportion of all = probability of one

Pay special attention to the two interpretations.

331 “Standard normal random variable” simply means the z-score. It’s “standardized” because μ=0 and σ=1 for z-scores.

We won’t have to convert to z-scores in this chapter, but still a z-score is a good yardstick. 95% of population is within z=±2, so values outside that are unusual (5% of cases). Values outside z=±3 are quite unusual (3 cases per thousand).

332 note difference between X and Z variables; always label axis correctly

333 review concepts #1–3,6,23,29

7.3  Applications of the ND

★ The types of problems in this section are the most important in the chapter.

349 ★ Steps 2–4 are replaced by section 1 of Normal Calculations on TI-83/84 or TI-89.

Always draw and shade the normal curve. Label X axis.

351 ★ Try Example 3 with calculator. (Correct answer is 0.5365; see sidebar.) Always round probabilities to four decimal places.

Important points for your sketches:

1. Draw the axis line.
2. Label the axis, x or z as appropriate.
3. Draw a vertical line in the middle of the distribution and show the numerical value of the mean.
4. Draw a horizontal line at about the right spot and show the numerical value of the standard deviation.
5. Draw a line and show the value for each boundary.
   Don’t draw other lines or label other numbers.
6. Shade the area you’re trying to find.
7. (optional) After you find the area, show its value.

349–50 ★ Try Example 1 with calculator. (Correct answer is 0.1203.)

★ in-class example: Men’s heights μ=69.3″, σ=2.92″. Find and interpret P(x≤60″).

Caution: The negative power of 10 matters! You report the given probability of 7.24...E-4 as 7.24×10^-4 or (better) 0.0007.

352 ★ Steps 2–3 are replaced by section 1 of the handout.

Always draw and shade the normal curve. Label X axis and optionally Z.

★ Work Example 4 with calculator.

Your sketch is similar to the above, except (7) you start with an area number (in the shading or above it, not under the axis), and (5) the boundaries aren’t known initially so you use symbols such as x₁.

★ In-class example: heights for middle 95% of adult females, given μ = 64.1″, σ = 2.75″. Answer: 58.7″ to 69.5″

354 ★ Practice #4, 14.

7.2  Standard Normal Distribution

The standard ND is simply a ND with μ=0, σ=1. It’s a ND done with z scores instead of real-world data (x scores).

This section is not terribly important because it’s basically theory that your calculator handles behind the scenes. Spend more time on 7.3.

337 properties — compare to page 328.

338–40 Work examples 1, 3, and 4 on calculator; only ★ Example 1 done in class. (See section 1 of Normal Calculations on TI-83/84 or TI-89.)

Always draw the normal curve and label the axis correctly. Round probabilities to four decimals.

341–2 Work example 5 using calculator.

342 ★ See last paragraph — always think in those terms. This is one of many ways you test your answers for reasonableness.

343 Work example 7 using calculator.

343–4 Work example 8 using calculator.

Remember symmetry: when you know one z score, you know the other.

344 Notation z_α from box at bottom, also written z(α), can be confusing. Remember: z_something is a z score, not an area. The subscripted “something” is the area.

345 ★ Work Example 9 with calculator.

346–7 Practice with 1–4, 26

7.4  Assessing Normality

How do you know if a sample comes from a normal distribution? Either look at a Minitab plot like those in the book, or check it yourself using MATH200A part 4. Don’t have the program? Use Normality Check on TI-83/84.

357–8 Read this part for concept, not calculation details. FYI, MATH200A part 4 works by doing the calculations and plots in the purple box.

358–9 ★ Work Example 1 on your calculator.

Check n (upper left of screen). You have a ND if r is greater than the critical value shown. (More precisely, if r<CRIT you don’t have a ND, but if r>CRIT you merely can’t rule out normality.)

(The critical value is different from the Decision Points for Correlation Coefficient you studied in Chapter 4. If you have a classic TI-83, as opposed to Plus or Silver, your program won’t compute the critical value. Use the formula 1.0063−.6118/n+1.3505/n²−.1288/√n in that case.)

360–1 ★ Compare the plots to see how far off from linearity is too far.

362–3 ★ Practice with problem 10.

7.5  Normal Approximation to Binomial PD

Skip this section.

What’s New

• 12 Mar 2013: Reference the handout to use if you don’t have the program; lose the rule of thumb in favor of comparing r and CRIT; give the formula for CRIT.
• (intervening changes suppressed)
• 12 Mar 2008: new document (MS-Word format)