Source: ENGR 216 Comprehensive Exam Practice Bank
Tags: normal distribution, Gaussian, z-score, z-table, standard deviation, variance, Central Limit Theorem, Galton machine, confidence interval, population mean, sample mean, ENGR 216, Texas A&M
The normal (Gaussian) distribution is the workhorse of experimental statistics. Z-scores let you convert any normal distribution to the standard normal, making probability lookups straightforward. Confidence intervals use z-scores in reverse: given a sample, they tell you the range where the true population mean likely sits. The Central Limit Theorem explains why the normal distribution appears so often in practice.
Normal distribution (Gaussian distribution)
A symmetric, bell-shaped probability distribution fully described by its mean ($\mu$) and standard deviation ($\sigma$). Approximately 68% of data falls within $\pm 1\sigma$, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$.
Variance ($\sigma^2$)
The average of the squared deviations from the mean. The standard deviation is its square root.
Standard deviation ($\sigma$)
A measure of spread. For a normal distribution, it sets the width of the bell curve.
Z-score (standard score)
The number of standard deviations a data point lies from the mean:
$$z = \frac{x - \mu}{\sigma}$$
Standard normal distribution
A normal distribution with $\mu = 0$ and $\sigma = 1$. All z-table lookups reference this distribution.
Central Limit Theorem (CLT)
States that the sum (or average) of a large number of independent random variables will be approximately normally distributed, regardless of the original distribution of each variable.
Galton machine (quincunx / bean machine)
A physical demonstration of the CLT. A ball drops through rows of pegs, making a random left-or-right deflection at each peg. The sum of many small random movements produces a bell-shaped distribution at the bottom.
Confidence interval
A range of values, calculated from sample data, that is expected to contain the true population parameter with a specified probability (confidence level).
Confidence level
The probability that the interval contains the true parameter (e.g. 85%, 92%, 95%).
If every value in a distribution is transformed so that its distance from the mean is scaled by a constant factor c, then the standard deviation scales by the same factor.
Worked example (tulip stems):
Common population: $\mu = 58$ cm, variance $= 78$ cm², so $\sigma = \sqrt{78} = 8.8318$ cm.
Experimental population: every stem's distance from the mean is exactly 3 times what it would be in the common population. The standard deviation therefore becomes:
$$\sigma_{\text{exp}} = 3 \times \sqrt{78} = 3 \times 8.8318 = 26.50 \text{ cm}$$
To find a specific cutoff value given a probability:
Identify the cumulative probability from the problem statement.
Look up the corresponding z-value in the z-table.
Convert back to the original scale: $x = \mu + z\sigma$.
Worked example (air filter):
The filter lets through the smallest 10% of particles. This means we need the z-value where the cumulative probability from the left equals 0.10.
From the z-table: $z = -1.28$ (the value where $P(Z \leq z) = 0.10$).
Given $\mu = 0.5$ microns and $\sigma = 0.2$ microns:
$$x = 0.5 + (-1.28)(0.2) = 0.5 - 0.256 = 0.244$$
The largest particle passing through has a diameter of 0.244 microns.
Each peg gives the ball a small, random deflection (left or right with roughly equal probability). The ball's final horizontal position is the sum of all those independent random deflections across many rows. By the Central Limit Theorem, the sum of many small, independent random variables converges to a normal distribution. This is why the collected balls form a bell-shaped curve at the bottom.
The formula for a confidence interval when the population standard deviation ($\sigma$) is known:
$$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
where:
$\bar{x}$ is the sample mean
$z_{\alpha/2}$ is the critical z-value for the desired confidence level
$\sigma$ is the population standard deviation
$n$ is the sample size
The margin of error (half-width of the interval) is:
$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
Given: $n = 176$, $\bar{x} = 140$ mmHg, $\sigma = 34$ mmHg.
For an 85% confidence level, $\alpha = 0.15$, so $\alpha/2 = 0.075$. The z-value where the cumulative probability equals $0.075$ is $z = -1.44$ (equivalently, $z_{\alpha/2} = 1.44$).
$$\text{Lower bound} = 140 - 1.44 \times \frac{34}{\sqrt{176}}$$
$$= 140 - 1.44 \times 2.5625$$
$$= 140 - 3.69 = 136.31$$
Given: lower bound = 50, $\sigma = 4.4$, $n = 26$, confidence level = 92%.
For 92%, $\alpha = 0.08$, $\alpha/2 = 0.04$. The z-value for cumulative probability 0.04 is $z = -1.75$ (so $z_{\alpha/2} = 1.75$).
The lower bound formula is:
$$\text{Lower bound} = \bar{x} - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
Solve for $\bar{x}$:
$$\bar{x} = 50 + 1.75 \times \frac{4.4}{\sqrt{26}} = 50 + 1.75 \times 0.8630 = 50 + 1.51 = 51.51$$
Rounded to zero decimal places: $\bar{x} = 52$.
Formula | Use |
|---|---|
$z = \frac{x - \mu}{\sigma}$ | Convert a value to a z-score |
$x = \mu + z\sigma$ | Convert a z-score back to original units |
$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ | Confidence interval (known $\sigma$) |
$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ | Margin of error |
Common z-values for confidence levels:
Confidence Level | $z_{\alpha/2}$ |
|---|---|
85% | 1.44 |
90% | 1.645 |
92% | 1.75 |
95% | 1.96 |
99% | 2.576 |
⚠️ "Variance" and "standard deviation" are tested interchangeably. If given variance, take the square root before doing anything else.
⚠️ Z-table direction matters. "Smallest 10% pass through" means cumulative probability = 0.10 from the left, giving a negative z-value.
⚠️ When a problem says the distance from the mean is scaled by a factor, that factor applies directly to $\sigma$, not to $\sigma^2$.
⚠️ For confidence intervals, $\alpha = 1 - \text{confidence level}$. Split $\alpha$ in half for a two-sided interval. A common exam slip is forgetting to halve $\alpha$.
⚠️ The CLT explains the Galton machine. The exam answer is concise: the sum of many independent random variables converges to a normal distribution.
⚠️ When solving for $\bar{x}$ from a lower bound, you add the margin of error (lower bound + E), not subtract it.
Q: Common tulip stems have a mean of 58 cm and variance of 78 cm². In an experimental population, every stem's distance from the mean is 3 times the common distance. What is the experimental standard deviation?
A: $3 \times \sqrt{78} = 3 \times 8.8318 = 26.50$ cm.
Q: Particle diameters are normally distributed with $\mu = 0.5$ and $\sigma = 0.2$ microns. A filter captures the largest 90%. What is the diameter of the largest particle that passes through?
A: $x = 0.5 + (-1.28)(0.2) = 0.244$ microns. The 10th percentile from the z-table gives $z = -1.28$.
Q: Based on the Galton machine, why does the sum of many small random movements produce a bell curve?
A: The Central Limit Theorem. The sum of many independent random values (left/right deflections) converges to a normal distribution regardless of the distribution of each individual deflection.
Q: Sample of 176, mean 140 mmHg, $\sigma = 34$ mmHg. What is the lower bound of an 85% confidence interval?
A: $140 - 1.44 \times (34/\sqrt{176}) = 140 - 3.69 = 136.31$.
Q: A 92% confidence interval has a lower bound of 50, $\sigma = 4.4$, and $n = 26$. What is the sample mean?
A: $\bar{x} = 50 + 1.75 \times (4.4/\sqrt{26}) = 52$ (rounded to zero decimal places).
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