B. Probability Distributions · Continuous Distributions and the Normal Distribution

Continuous Distributions and the Normal Distribution: Unit Review

B2.1 B2.2 B2.3 B2.4 B2.5 B2.6 B2.7 B2.8About 40 minutes

Learning goals

I can tell discrete from continuous variables, measure spread with standard deviation, and organize continuous data.
I can explain density curves and find probabilities as areas, including for the normal distribution.
I can use z-scores, the z-table, the empirical rule and the normal approximation to the binomial to solve problems.

Words to know

density curve
A curve with total area 1 that models a continuous variable; probabilities are areas under it.

normal distribution N(μ, σ²)
The symmetric bell-shaped distribution with mean μ and standard deviation σ.

z-score
z = (x − μ)/σ, the number of standard deviations from the mean.

continuity correction
The ±0.5 adjustment used when a normal curve approximates a discrete distribution.

Let's review!

In this unit you separated continuous random variables (measured) from discrete ones (counted). You measured spread with the standard deviation, including for grouped data, organized data in histograms and frequency polygons, and saw how they lead to smooth density curves where probability is area and single values have probability 0. Then you met the normal distribution, the empirical rule, z-scores and the z-table, worked backwards to percentiles, saw the binomial and hypergeometric distributions turn normal, and used the normal approximation with a continuity correction. Finally, you used all of it in applications.

Next in this course: the units Organization of Data for Analysis and One-Variable Data Analysis use these ideas with real data sets. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time with a calculator. For every normal question, sketch the curve, mark μ and the value, and shade the region before calculating. Round z-scores to 2 decimal places for the table; keep other values unrounded until the end. Use the table of values below, or the full table in the lesson Using the z-Table to Find Normal Probabilities. For each question you miss, write down why (wrong formula? wrong tail? forgot ±0.5?) and go back to that lesson. Finish with the 10-question quiz.

How the unit fits together. Lesson numbers are in brackets. Continuous data lead to density curves and the normal model; z-scores and the table turn the model into probabilities; the binomial joins in through the normal approximation.

z	P(Z < z)	P(Z < −z) = 1 − P(Z < z)
0.50	0.6915	0.3085
0.67	0.7486	0.2514
0.84	0.7995	0.2005
0.99	0.8389	0.1611
1.28	0.8997	0.1003
1.50	0.9332	0.0668
1.64	0.9495	0.0505
1.65	0.9505	0.0495
1.67	0.9525	0.0475
2.33	0.9901	0.0099
3.10	0.9990	0.0010

z-table values used in this review (from the full table in the lesson Using the z-Table to Find Normal Probabilities). The middle column is the area to the left of z; the last column is the area to the left of −z.

Discrete and Continuous Random Variables

Discrete or continuous: (a) the number of moose seen on a drive; (b) the time of the drive?

A mass is recorded as 5.3 kg to the nearest 0.1 kg. What interval of true masses does it represent?

Why do bars touch in a histogram of continuous data but not in a bar graph of counts?

Standard Deviation: Measuring Spread

Find the sample standard deviation of 4, 6, 8, 10.

Find the population standard deviation of the same four values.

If 5 is added to every value, what happens to the SD? If every value is tripled?

Mean and Standard Deviation of Grouped Continuous Data

Classes 0 to 10, 10 to 20, 20 to 30 have frequencies 2, 5, 3. Estimate the mean.

For the same table (a population), estimate σ.

Why are grouped results only estimates?

Frequency Tables, Histograms and Frequency Polygons

A class holds 6 of 24 values. What is its relative frequency?

Classes 10 to 20 (4), 20 to 30 (9), 30 to 40 (2). List the frequency polygon points.

What goes wrong if the class width is much too small for 30 values?

From Frequency Polygons to Density Curves

A class has relative frequency 0.2 and width 4. What is its relative frequency density?

Find k so that f(x) = k for 0 ≤ x ≤ 25 is a density.

Give two reasons continuous distributions need mathematical models.

Probability as Area

X is uniform from 0 to 30. Find P(X > 18).

For the same X, find P(X = 10).

f(x) = x/18 for 0 ≤ x ≤ 6. Find P(X < 3).

For a continuous X, is P(X ≤ 4) different from P(X < 4)?

The Normal Distribution and Its Properties

For N(80, 6²), where are the inflection points?

In a normal distribution, how are the mean and median related?

Which is wider and flatter: N(50, 4²) or N(50, 9²)?

The Empirical Rule: 68, 95, 99.7

X ~ N(80, 6²). What percent is between 68 and 86?

X ~ N(80, 6²). What percent is above 92?

In a group of 400 modelled by N(80, 6²), about how many are below 74?

z-Scores and the Standard Normal Distribution

X ~ N(80, 6²). Find the z-score of 71.

X ~ N(80, 6²). What value has z = 2.5?

Ana scored 72 (μ = 64, σ = 5); Ben scored 81 (μ = 75, σ = 4). Who did relatively better?

Using the z-Table to Find Normal Probabilities

Find P(Z < −0.84).

Find P(0.5 < Z < 1.5).

X ~ N(80, 6²). Find P(X > 89).

Working Backwards: Percentiles, Cut-offs and Unknown Parameters

X ~ N(80, 6²). Find the 90th percentile.

X ~ N(80, 6²). Find Q1 and Q3.

σ = 6 and only 5% of values may be below 70. What must μ be?

How Binomial and Hypergeometric Distributions Become Normal

X ~ Binomial(60, 0.3). Find μ and σ.

Is a normal model reasonable for Binomial(60, 0.3)?

What is the total area of any probability histogram?

The Normal Approximation to the Binomial and Hypergeometric Distributions

Continuity-correct P(X ≥ 22) and P(X < 22).

X ~ Binomial(60, 0.3). Estimate P(X ≥ 22).

Why is the continuity correction needed?

Applications: Solving Problems with Normal Models

Bags are N(505, 3²) g. What proportion are under 500 g?

A part's lifetime is N(1000, 50²) h. What warranty makes at most 1% fail within it?

A measurement has z = −3.1 under a claimed model. What do you conclude?

Just for fun

Why did the normal curve get invited to every party? Because it gets along with 68% of people right away, 95% after a little effort, and 99.7% of everyone eventually.

One picture to remember: the standard normal curve, the z-score formula and the empirical rule.

Check your understanding

Quiz

Which random variable is continuous?

A sample is 2, 4, 6. What is s?

For a continuous random variable X, which is always true?

X ~ N(60, 10²). About what percent of values lie between 40 and 70?

X ~ N(60, 10²). What is the z-score of 47?

What is P(Z > 1.5)?

Which z-score is the 90th percentile of Z?

A normal curve has inflection points at 30 and 50. What are μ and σ?

X ~ Binomial(100, 0.5). Which normal model approximates it?

10.

Using the normal approximation, which interval approximates P(X = 30)?

For parents and teachers

Use the review to find what to revisit:

Calling measured quantities discrete, or wrong measurement intervals → Lesson 1 (Discrete and Continuous Random Variables).
Standard deviation errors (forgetting to square, the square root, or n − 1) → Lesson 2 (Standard Deviation).
Grouped mean or SD wrong (dividing by the number of classes, using boundaries) → Lesson 3 (Grouped Data).
Overlapping classes, polygon points at boundaries, poor class widths → Lesson 4 (Histograms and Frequency Polygons).
Treating curve height as probability, or density not having area 1 → Lesson 5 (Density Curves).
Thinking P(X = a) > 0 for continuous X, or triangle-area slips → Lesson 6 (Probability as Area).
Muddled properties of the normal curve, or reading N(μ, σ²) wrongly → Lesson 7 (The Normal Distribution).
Using 68% for one side of the mean → Lesson 8 (The Empirical Rule).
Sign or order errors in z = (x − μ)/σ → Lesson 9 (z-Scores).
Left/right tail confusion, negative z lookups → Lesson 10 (Using the z-Table).
Looking up areas in the z column, or using 1.96 for the top 5% → Lesson 11 (Working Backwards).
Using npq as σ, or skipping the np ≥ 5 check → Lesson 12 (Binomial and Hypergeometric Become Normal).
Continuity correction in the wrong direction → Lesson 13 (The Normal Approximation).
Choosing the wrong model, or no assumptions stated → Lesson 14 (Applications).

Looking ahead: the units Organization of Data for Analysis and One-Variable Data Analysis apply z-scores, percentiles and the normal model to real data, and the Culminating Data Management Investigation is a chance to use them in a project of your teen's own choosing.