B. Probability Distributions · Continuous Distributions and the Normal Distribution

Applications: Solving Problems with Normal Models

B2.6 B2.7 B2.8About 75 minutes

Learning goals

I can choose a suitable model (normal, binomial, or a normal approximation) for a real situation and justify it.
I can solve multi-step problems that combine normal probabilities with other probability tools.
I can use normal probabilities to make decisions, such as setting tolerances and warranties or judging whether a result is unusual.
I can communicate answers clearly, in context, and state the assumptions I made.

Words to know

tolerance (specification limits)
The acceptable range for a measurement, such as 9.97 mm to 10.03 mm for a bolt's diameter.

quality control
Using measurements and probability to keep a product within its specifications.

assumption
A condition you accept in order to use a model, such as "the masses are normally distributed" or "the trials are independent".

warranty period
The time during which a company promises to replace a product that fails.

Normal models at work

Normal distributions are used every day by quality engineers at food and auto-parts plants, by actuaries setting insurance prices, by pharmacists and pharmacologists checking dose amounts, by foresters planning tree nurseries, and by data analysts in almost every field. In university, statistics courses in science, business, psychology, kinesiology and engineering all build on what you have learned in this unit.

Real problems rarely say "find P(Z < 1.27)". You have to decide which model fits, check its conditions, and explain what the answer means. This lesson pulls the whole unit together.

A five-step routine for any probability problem about a continuous variable or a large count.

z	P(Z < z)	P(Z < −z) = 1 − P(Z < z)
0.33	0.6293	0.3707
0.60	0.7257	0.2743
1.04	0.8508	0.1492
1.27	0.8980	0.1020
1.28	0.8997	0.1003
1.33	0.9082	0.0918
1.50	0.9332	0.0668
1.64	0.9495	0.0505
1.65	0.9505	0.0495
1.67	0.9525	0.0475
1.88	0.9699	0.0301
2.00	0.9772	0.0228
2.05	0.9798	0.0202
2.24	0.9875	0.0125
2.33	0.9901	0.0099
2.71	0.9966	0.0034
2.80	0.9974	0.0026

z-table values used in this lesson (taken from the full standard normal 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.

Worked example

Example 1: Normal, then binomial

Seedling heights are N(32, 10.2²) cm. A seedling taller than 45 cm is sold as a "specimen tree". (a) What proportion of seedlings are specimen trees? (b) A customer picks 5 seedlings at random. What is the probability that at least 2 are specimen trees?

(a) z = (45 − 32)/10.2 ≈ 1.27. P(X > 45) = 1 − 0.8980 = 0.1020.
(b) Now each seedling is a trial with p = 0.102. Let Y be the number of specimen trees among 5; Y ~ Binomial(5, 0.102). (np is only 0.51, so use the binomial formula, not a normal approximation.)
P(Y = 0) = 0.898⁵ ≈ 0.5840.
P(Y = 1) = 5(0.102)(0.898)⁴ ≈ 0.3316.
P(Y ≥ 2) = 1 − 0.5840 − 0.3316 ≈ 0.0844.

(a) 0.1020 (about 10%) (b) ≈ 0.084 (about 8%).

Worked example

Example 2: Comparing two machines

Two machines fill 500 g bags of flour. Machine A: N(505, 3²). Machine B: N(510, 6²). Which machine produces fewer underweight bags (under 500 g)? Which would you recommend?

A: z = (500 − 505)/3 ≈ −1.67. P = 1 − 0.9525 = 0.0475.
B: z = (500 − 510)/6 ≈ −1.67. P = 0.0475.
Both machines underfill about 4.75% of bags: the same z-score gives the same tail area.
But A's mean is 5 g lower, so A gives away 5 g less flour per bag on average, and its smaller σ makes it more consistent.

Both produce about 4.75% underweight bags. Recommend Machine A: same underweight rate, less flour given away, more consistent bags.

The two flour machines. Machine B's curve (dashed) is centred higher but spread wider, so its tail below 500 g has the same area as Machine A's.

Worked example

Example 3: Is this result unusual?

A brand's 250 g chocolate bars have masses N(252, 5²) g. You buy one and it weighs 238 g. Is that unusual enough to complain?

z = (238 − 252)/5 = −2.80.
P(X < 238) = 1 − 0.9974 = 0.0026.
Only about 26 bars in 10 000 would be this light or lighter if the model is right. That is more than 2.5 standard deviations below the mean.

Yes, very unusual (probability ≈ 0.0026). Either you were unlucky, or something is wrong with the filling process. It is reasonable to contact the company.

Worked example

Example 4: Setting a warranty

Battery lives are N(400, 25²) hours. The company will replace batteries that fail during the warranty period, and it wants to replace no more than 1% of them. How long should the warranty be?

We need the 1st percentile. In the table, P(Z < 2.33) = 0.9901, so the 1st percentile is z ≈ −2.33.
x = 400 + (−2.33)(25) = 400 − 58.25 = 341.75.
Round down so that the percentage replaced stays at or below 1%.

A warranty of about 341 hours.

State your assumptions

Every model rests on assumptions. Say them out loud in your answers:

"Assuming the heights are normally distributed ..." (Check with a histogram or the 68–95–99.7 counts.)
"Assuming the seedlings are chosen independently ..." (needed for the binomial in Example 1).
"Assuming np ≥ 5 and nq ≥ 5 ..." (needed for a normal approximation).

If the data are clearly skewed (mean far from the median, a long tail), a normal model can give badly wrong answers in the tails. That is where many real decisions are made, so check first. You will check data more carefully in the unit One-Variable Data Analysis.

Just for fun

Quality-control joke: a statistician's bag of chips said 200 g and weighed 199 g. She did not complain. She calculated the z-score, shrugged, and ate the chips with confidence.

Hands-on activity

A kitchen quality-control study

You need: a kitchen scale that reads to 1 g (or better), 20 similar items (crackers, cookies, apples, eggs or slices of bread), a calculator, paper. Ask a grown-up before using food, and eat or store it afterwards.

Weigh each of the 20 items and record the masses.
Find x̄, s and the median. Draw a histogram (about 5 classes).
Decide whether a normal model looks reasonable (shape, mean close to median, about 68% within 1 SD).
Using N(x̄, s²), predict the proportion of items lighter than x̄ − 1.5s, and compare with your count.
Write two sentences, as a quality-control report, about how consistent the product is.

What to look for: about 6.7% of items (1 or 2 of 20) should be lighter than x̄ − 1.5s if the masses are normal. With only 20 items, 0 to 3 is normal variation. A clear skew or two separate clumps means the normal model does not fit.

Your turn

Tire tread lives are N(70 000, 8000²) km. What percent of tires last more than 85 000 km?

For the tires, the company wants only 2% to wear out before the warranty distance. What distance should the warranty be?

Bolt diameters are N(10.00, 0.02²) mm. The specification is 9.97 to 10.03 mm. What percent of bolts meet it?

Using your last answer, what is the probability that all 4 bolts in a package meet the specification? What assumption do you need?

A commute is N(42, 7²) min. Today it took 61 min. Is that unusual?

A factory makes 500 items a day, each defective with probability 0.03, independently. Which model describes the daily number of defects? Can a normal approximation be used? Give μ and σ.

A data set of wait times has mean 12 min, median 9 min and maximum 40 min. Should you use a normal model to find the proportion waiting over 30 min? Explain.

More practice: warm-up

What does it mean to "state your assumptions"?

X ~ N(50, 2²). Find the z-score of 47.

Using the table, find P(X < 47) for X ~ N(50, 2²).

Is a value with z = 0.4 unusual?

Name one career that uses normal distributions.

More practice: on your own

Juice machine P fills 1 L cartons with N(1004, 2²) mL; machine Q with N(1008, 4²) mL. Compare the percent under 1000 mL. Which machine would you choose?

Marks are N(66, 12²). The top 15% get an A. What is the lowest A mark?

A nursery sells seedlings in trays of 10. Using P(height > 45 cm) = 0.102 from Example 1, find the expected number of specimen trees in a tray.

A ride requires riders to be at least 120 cm tall. Suppose heights of 8-year-olds are modelled by N(128, 6²) (an example model). What percent of 8-year-olds are too short?

A company claims its bars weigh N(252, 5²) g. A store inspector weighs one bar: 249 g. Is that evidence against the claim? Explain.

Challenge

A bakery's bags come 60% from machine A, N(505, 3²) g, and 40% from machine B, N(510, 6²) g. What proportion of all bags weigh more than 512 g?

Tire lives are normal. 5% of tires wear out before 55 000 km and 10% last longer than 82 000 km. Find μ and σ.

A company claims that 90% of its customers are satisfied. In a random survey of 200 customers, only 170 say they are satisfied. If the claim were true, how likely is a result of 170 or fewer? What do you conclude?

Check your understanding

Quiz

Heights are N(32, 10.2²) cm. Which calculation gives the proportion above 45 cm?

Two machines have the same z-score for the 500 g limit. What does that tell you?

A value has z = −2.8 under a claimed normal model. What is the best conclusion?

Battery lives are N(400, 25²) h. A warranty should cover failures for at most 1% of batteries. About how long should it be?

Which situation is best modelled by a binomial distribution rather than a normal one?

Which assumption is needed to say P(all 4 bolts meet spec) = p⁴?

For parents and teachers

Big idea (B2.6, B2.7, B2.8): real problems need a model choice (normal for measurements, binomial for counts, a normal approximation for large counts), a sketch, a calculation, and an answer in context with its assumptions.

How to help: The kitchen study is a great way to finish the unit. Ask your teen to present their "quality-control report" to you as if you were the factory manager: Is the product consistent? Is a normal model reasonable? What proportion is underweight?

Common difficulties:

Forgetting to switch models in multi-step problems (normal first, then binomial).
Rounding warranties or cut-offs the wrong way. Think about which side keeps the promise.
Saying "unusual" without a probability or z-score to back it up.
Using a normal model for clearly skewed data.