Left area: shorter than 174 cm

Women's heights in a country are normally distributed with $\mu = 165$ cm and $\sigma = 6$ cm. What proportion of women are shorter than 174 cm?

Sketch and shade: Shade the area to the left of 174 cm, which lies to the right of the mean 165 cm = The answer should be more than 0.5

Standardize: $z = \frac{174 - 165}{6} = \frac{9}{6}$ = $z = 1.5$

Read the z-table: Row 1.5, column 0.00 = $P(Z < 1.5) = 0.9332$

Interpret: A left area needs no adjustment = $P(X < 174) = 0.9332$

Answer: $P(X < 174) = 0.9332$: about 93.32% of women are shorter than 174 cm.

Right area: batteries lasting over 14 hours

A phone battery lasts $\mu = 12$ hours on average with $\sigma = 1.6$ hours, normally distributed. What is the probability that a battery lasts more than 14 hours?

Sketch and shade: Shade the right tail beyond 14 hours = A small area, less than 0.5

Standardize: $z = \frac{14 - 12}{1.6} = \frac{2}{1.6}$ = $z = 1.25$

Read the z-table: Row 1.2, column 0.05 = $P(Z < 1.25) = 0.8944$

Take the complement: The table gives the left area, so subtract from 1: $1 - 0.8944$ = $P(X > 14) = 0.1056$

Answer: $P(X > 14) = 0.1056$: about 10.56% of batteries last longer than 14 hours.

Area between two values: exam scores

Exam scores are normally distributed with $\mu = 70$ and $\sigma = 8$. What percentage of students score between 62 and 86?

Standardize the lower value: $z_1 = \frac{62 - 70}{8} = \frac{-8}{8}$ = $z_1 = -1$

Standardize the upper value: $z_2 = \frac{86 - 70}{8} = \frac{16}{8}$ = $z_2 = 2$

Read both left areas: $P(Z < 2) = 0.9772$ and $P(Z < -1) = 0.1587$ = Two table values

Subtract: big left area minus small left area: $0.9772 - 0.1587$ = $0.8185$

Answer: $P(62 < X < 86) = 0.8185$, so about 81.85% of students score between 62 and 86.

Inverse problem: the score for the top 10%

With the same exam ($\mu = 70$, $\sigma = 8$), the top 10% of students receive a certificate. What is the lowest score that earns a certificate?

Turn the top 10% into a left area: Top 10% to the right means $1 - 0.10 = 0.90$ to the left = Look for 0.90 inside the table

Find z in the table: The closest table value is $0.8997$, in row 1.2 and column 0.08 = $z \approx 1.28$

Go back to real units: $x = \mu + z\sigma = 70 + 1.28 \times 8 = 70 + 10.24$ = $x = 80.24$

Answer: A score of about 80.24 or more is in the top 10%. With whole points, a student needs 81.

From probability to a count: underfilled bottles

A machine fills bottles with $\mu = 500$ mL and $\sigma = 4$ mL. It fills 2000 bottles a day. About how many bottles contain less than 494 mL?

Standardize: $z = \frac{494 - 500}{4} = \frac{-6}{4}$ = $z = -1.5$

Read the z-table: Row -1.5, column 0.00 = $P(Z < -1.5) = 0.0668$

Multiply by the number of bottles: $0.0668 \times 2000$ = $133.6$

Answer: About 134 bottles a day (6.68% of 2000) contain less than 494 mL.

Mistake: Using the table value directly for a "greater than" question, for example $P(X > 14) = 0.8944$

Why: The z-table always gives the area to the LEFT of $z$. A right tail is everything else.

Correct: Subtract from 1: $P(X > 14) = 1 - 0.8944 = 0.1056$. A quick sketch shows the right tail is small, so 0.8944 cannot be right.

Mistake: Subtracting the z-scores first: $P(-1 < Z < 2) = P(Z < 2 - (-1)) = P(Z < 3)$

Why: Areas under a curve do not work like that. You need the area up to each bound, then the difference between those areas.

Correct: Look up both left areas and subtract them: $0.9772 - 0.1587 = 0.8185$.

Mistake: Standardizing in the wrong order or dividing by the wrong number: $z = \frac{\mu - x}{\sigma}$ or $z = \frac{x - \mu}{\sigma^2}$

Why: Reversing the subtraction flips the sign, so you land on the wrong side of the mean. Dividing by the variance gives the wrong distance.

Correct: Always $z = \frac{x - \mu}{\sigma}$: values above the mean get a positive $z$, values below get a negative $z$.

Mistake: In inverse problems, using the percentage as the z-score (z = 0.90) or using the wrong tail (z = -1.28 for the top 10%)

Why: The percentage is an AREA, which you look for inside the table. The z-score is read from the row and column edges.

Correct: Top 10% means 0.90 to the left, which gives $z \approx 1.28$ (positive, above the mean). Bottom 10% gives $z \approx -1.28$.

Quality Control: Underfilled Cans

Drink factories check what share of cans fall below the amount printed on the label, and adjust the filling machines.

A line fills cans with $\mu = 330$ mL and $\sigma = 2$ mL. Any can below 327 mL is flagged as underfilled.

Growth Charts at the Doctor

Pediatricians compare babies with growth charts. Each curve on the chart is a percentile of a normal distribution.

Weights of 6-month-old babies in a study are about normal with $\mu = 8$ kg and $\sigma = 0.5$ kg. A baby below the 10th percentile gets an extra check-up.

Planning a City Marathon

Race organizers estimate how many runners will be on the course at certain times, to plan water stations and staff.

Finishing times are about normal with $\mu = 240$ minutes and $\sigma = 30$ minutes. Organizers want to know the share of runners finishing between 225 and 285 minutes.

Q: Is $P(X < 174)$ different from $P(X \le 174)$?

A: No. For a continuous variable a single value has probability 0, so $<$ and $\le$ give the same area. This is different from the binomial distribution, where it matters.

Q: My z-score has three decimals, like 1.645. What do I do?

A: Round to two decimals for the table, or take a value between the two neighbouring entries. For $z = 1.645$ the areas 0.9495 and 0.9505 are equally close, so 0.95 is a good answer.

Q: What if the z-score is beyond the table, like $z = 4$?

A: Tables usually stop at about 3.49. Beyond that, the left area is practically 1 (for large positive z) or practically 0 (for large negative z). Such values are extremely rare.

Q: Why does the percentile give the area to the left?

A: The 90th percentile is the value with 90% of the population below it. "Below" is the left side of the curve, which is exactly what the z-table measures.