Q: What is inverse problems (percentiles)?

To find the value $x$ such that $P(X \le x) = p$, find the corresponding $z$ from a table or invNorm, then transform: $x = \mu + z \sigma$. The 90th percentile of $Z$ is $z \approx 1.28$, the 95th is $z \approx 1.645$, the 97.5th is $z \approx 1.96$.

Q: What is direct use of the empirical rule?

Heights of adult males in a city are normally distributed with $\mu = 175$ cm and $\sigma = 7$ cm. About what percentage of men are taller than $189$ cm?

Q: What is two-sided interval?

For the same distribution, what percentage are between $168$ and $182$ cm?

Q: What is mixed empirical-rule interval?

For $X \sim N(50, 100)$ (so $\sigma = 10$), find $P(30 < X < 60)$.

Q: What is comparing two distributions with z-scores?

Two students sit different tests. Alex scores $82$ on a test with $\mu = 70$, $\sigma = 8$. Sam scores $75$ on a test with $\mu = 60$, $\sigma = 10$. Who performed better relative to their cohort?

Q: What is inverse normal?

For $X \sim N(100, 225)$ (so $\sigma = 15$), find the value below which $95\%$ of the data lies.

Question 1

What is the normal distribution?

Accepted Answer

A continuous random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, written $X \sim N(\mu, \sigma^2)$, if its pdf is

Question 2

What is z-scores?

Accepted Answer

The z-score of a value $x$ measures how many standard deviations it is from the mean:

Question 3

What is the empirical rule (68-95-99.7)?

Accepted Answer

By symmetry, $P(0 \le Z \le 1) \approx 0.34$, $P(0 \le Z \le 2) \approx 0.475$, $P(0 \le Z \le 3) \approx 0.4985$.

Question 4

What is computing probabilities?

Accepted Answer

For $X \sim N(\mu, \sigma^2)$ and $a < b$:

Question 5

What is inverse problems (percentiles)?

Accepted Answer

To find the value $x$ such that $P(X \le x) = p$, find the corresponding $z$ from a table or invNorm, then transform: $x = \mu + z \sigma$. The 90th percentile of $Z$ is $z \approx 1.28$, the 95th is $z \approx 1.645$, the 97.5th is $z \approx 1.96$.

Question 6

What is direct use of the empirical rule?

Accepted Answer

Heights of adult males in a city are normally distributed with $\mu = 175$ cm and $\sigma = 7$ cm. About what percentage of men are taller than $189$ cm?

Question 7

What is two-sided interval?

Accepted Answer

For the same distribution, what percentage are between $168$ and $182$ cm?

Question 8

What is mixed empirical-rule interval?

Accepted Answer

For $X \sim N(50, 100)$ (so $\sigma = 10$), find $P(30 < X < 60)$.

Question 9

What is comparing two distributions with z-scores?

Accepted Answer

Two students sit different tests. Alex scores $82$ on a test with $\mu = 70$, $\sigma = 8$. Sam scores $75$ on a test with $\mu = 60$, $\sigma = 10$. Who performed better relative to their cohort?

Question 10

What is inverse normal?

Accepted Answer

For $X \sim N(100, 225)$ (so $\sigma = 15$), find the value below which $95\%$ of the data lies.

Question 11

What is standardising with the wrong sign?

Accepted Answer

$z = \frac{x - \mu}{\sigma}$. A value below the mean has a negative z-score. Do not drop the sign.

Question 12

What is confusing $\sigma$ and $\sigma^2$?

Accepted Answer

$N(\mu, \sigma^2)$ uses the variance, but the empirical rule and z-score use $\sigma$. If a question gives $\sigma^2 = 25$, then $\sigma = 5$.

Question 13

What is forgetting symmetry?

Accepted Answer

$P(Z \ge -1) = P(Z \le 1) \approx 0.84$. Use the symmetry of the bell curve rather than computing tails twice.

Question 14

What is adding empirical rule pieces incorrectly?

Accepted Answer

$P(-1 \le Z \le 1) \approx 0.68$ is for the full two-sided interval. The one-sided half is $\frac{0.68}{2} = 0.34$. Do not double-count the central area.

Question 15

What is applying the empirical rule to non-normal data?

Accepted Answer

The $68$-$95$-$99.7$ rule is specific to the normal distribution. For other shapes you must use other methods.