Quiz: Probability Distributions and the Normal Curve
Test your understanding of probability distributions, the binomial and normal distributions, z-scores, and normality assessment. Try to answer each question before revealing the answer.
1. Which of the following is a discrete random variable?
(a) The weight of a randomly selected apple (b) The time it takes a student to finish an exam (c) The number of heads in 20 coin flips (d) The temperature outside at noon
Answer
**(c) The number of heads in 20 coin flips.** A discrete random variable takes on countable values. The number of heads can be 0, 1, 2, ..., 20 — a finite, countable set. The other options (weight, time, temperature) can take any value within a range, making them continuous.2. For a valid discrete probability distribution, which of the following must be true?
(a) All probabilities must be equal (b) All probabilities must be between 0 and 1, and they must sum to 1 (c) The expected value must be one of the possible outcomes (d) The distribution must be symmetric
Answer
**(b) All probabilities must be between 0 and 1, and they must sum to 1.** Probabilities don't need to be equal (that would only be the uniform distribution). The expected value doesn't have to be a possible outcome (e.g., $E(X) = 3.5$ for a die). The distribution doesn't need to be symmetric.3. The expected value of a random variable represents:
(a) The most likely outcome on a single trial (b) The median of the distribution (c) The long-run average over many trials (d) The value that occurs exactly 50% of the time
Answer
**(c) The long-run average over many trials.** The expected value $E(X)$ is the theoretical average you'd approach if you repeated the random process infinitely many times. It's not necessarily the most likely outcome (that's the mode), the median, or the 50% mark.4. Which of the following scenarios does NOT meet the conditions for a binomial distribution?
(a) Flipping a fair coin 30 times and counting heads (b) Rolling a die 20 times and counting the number of sixes (c) Drawing 10 cards from a deck without replacement and counting aces (d) Surveying 50 randomly selected voters and counting those who support a candidate
Answer
**(c) Drawing 10 cards from a deck without replacement and counting aces.** Drawing without replacement violates the independence condition. After drawing the first card, the probabilities change for the remaining cards. The other scenarios meet all four BINS conditions (Binary outcomes, Independent trials, Number fixed, Same probability).5. For a binomial distribution with $n = 20$ and $p = 0.3$, the expected value is:
(a) 3 (b) 6 (c) 10 (d) 14
Answer
**(b) 6.** $E(X) = np = 20 \times 0.3 = 6$. On average, you'd expect 6 successes out of 20 trials when each trial has a 30% chance of success.6. For a continuous random variable, $P(X = 5)$ equals:
(a) The height of the PDF at $x = 5$ (b) 0 (c) The area under the PDF to the left of 5 (d) 1 minus the area to the left of 5
Answer
**(b) 0.** For a continuous random variable, the probability of any single exact value is zero. This is because there are infinitely many possible values, so each individual point has probability zero. With continuous variables, we can only calculate probabilities for *ranges* of values (areas under the curve).7. The height of the probability density function (PDF) at a particular point represents:
(a) The probability of that exact value occurring (b) The cumulative probability up to that point (c) The relative likelihood of values near that point (density, not probability) (d) The expected value of the distribution
Answer
**(c) The relative likelihood of values near that point (density, not probability).** The PDF height tells you the relative density — how concentrated probability is around that value compared to other values. Only the *area* under the curve gives actual probability. A higher PDF value means values in that neighborhood are more likely than values where the PDF is lower.8. Which of the following is NOT a property of the normal distribution?
(a) It is symmetric about the mean (b) The mean, median, and mode are all equal (c) Approximately 68% of values fall within 1 standard deviation of the mean (d) The curve touches the x-axis at 3 standard deviations from the mean
Answer
**(d) The curve touches the x-axis at 3 standard deviations from the mean.** The normal curve *never* touches the x-axis — the tails extend to positive and negative infinity, approaching zero but never reaching it. While 99.7% of values fall within 3 standard deviations, the remaining 0.3% is in those thin tails beyond $\pm 3\sigma$.9. The standard normal distribution has:
(a) $\mu = 1$ and $\sigma = 0$ (b) $\mu = 0$ and $\sigma = 1$ (c) $\mu = 0$ and $\sigma = 0$ (d) $\mu = 100$ and $\sigma = 15$
Answer
**(b) $\mu = 0$ and $\sigma = 1$.** The standard normal distribution is the special case with mean 0 and standard deviation 1. Any normal distribution can be transformed to the standard normal using the z-score formula $z = (x - \mu) / \sigma$.10. A z-score of $z = -2.0$ means:
(a) The value is 2 standard deviations below the mean (b) The value is at the 2nd percentile (c) The probability of this value is 0.02 (d) The value occurs 2% of the time
Answer
**(a) The value is 2 standard deviations below the mean.** A z-score tells you how many standard deviations a value is from the mean. Negative z-scores are below the mean; positive ones are above. $z = -2.0$ means exactly 2 standard deviations below. The cumulative probability is about 0.0228 (2.28th percentile), but the z-score itself is a measure of *position*, not probability.11. If $X \sim N(500, 100^2)$, what is $P(X < 500)$?
(a) 0 (b) 0.25 (c) 0.50 (d) 1.00
Answer
**(c) 0.50.** Because the normal distribution is perfectly symmetric about the mean, exactly half the area is below the mean and half is above. For any $X \sim N(\mu, \sigma^2)$, $P(X < \mu) = 0.50$.12. Using the z-table, $P(Z \leq 1.00) = 0.8413$. What is $P(Z > 1.00)$?
(a) 0.8413 (b) 0.1587 (c) 0.3174 (d) 0.0228
Answer
**(b) 0.1587.** Since the total area under the curve is 1: $P(Z > 1.00) = 1 - P(Z \leq 1.00) = 1 - 0.8413 = 0.1587$. About 15.87% of the standard normal distribution falls above $z = 1.00$.13. Exam scores are normally distributed with $\mu = 75$ and $\sigma = 8$. What proportion of students scored between 67 and 83?
(a) About 50% (b) About 68% (c) About 95% (d) About 99.7%
Answer
**(b) About 68%.** $67 = 75 - 8 = \mu - 1\sigma$ and $83 = 75 + 8 = \mu + 1\sigma$. The range from $\mu - \sigma$ to $\mu + \sigma$ contains approximately 68% of a normal distribution (the Empirical Rule, now formalized). The exact value is 68.27%.14. You want to find the value $x$ such that $P(X \leq x) = 0.95$ for $X \sim N(200, 25^2)$. Which Python function would you use?
(a) stats.norm.cdf(0.95, 200, 25)
(b) stats.norm.ppf(0.95, 200, 25)
(c) stats.norm.pdf(0.95, 200, 25)
(d) stats.norm.pmf(0.95, 200, 25)
Answer
**(b) `stats.norm.ppf(0.95, 200, 25)`** The `ppf` function (percent point function) is the *inverse* of the CDF — it takes a probability and returns the corresponding value. `cdf` goes the other direction (value to probability). `pdf` gives the density (not probability). `pmf` is for discrete distributions (the normal is continuous, so it doesn't have a PMF).15. When using the normal approximation to the binomial, the continuity correction for $P(X \geq 8)$ changes the calculation to:
(a) $P(X \geq 7.5)$ (b) $P(X \geq 8.5)$ (c) $P(X > 8)$ (d) $P(X \geq 8) + 0.5$
Answer
**(a) $P(X \geq 7.5)$.** The continuity correction adjusts for the fact that the discrete value 8 "occupies" the space from 7.5 to 8.5 on the continuous scale. Since we want to *include* 8, we start from 7.5. If the question asked for $P(X > 8)$ (excluding 8), we'd use $P(X > 8.5)$.16. On a QQ-plot, if the points follow the diagonal reference line closely except that they curve upward at the right end, this suggests:
(a) The data is approximately normal (b) The data has a heavier right tail than a normal distribution (c) The data is left-skewed (d) The data has lighter tails than a normal distribution
Answer
**(b) The data has a heavier right tail than a normal distribution.** When the right end of a QQ-plot curves upward, it means the largest values in the data are *more extreme* than what a normal distribution would predict. This indicates a heavier right tail — the data has more extreme high values than expected. This is common with income data, for example.17. The Shapiro-Wilk test produces $W = 0.987$ and $p = 0.42$ for a sample of 50 observations. What do you conclude?
(a) The data is perfectly normal (b) There is strong evidence the data is not normal (c) There is not enough evidence to reject normality (d) The sample is too small for the test to be valid
Answer
**(c) There is not enough evidence to reject normality.** A large p-value (0.42 >> 0.05) means we fail to reject the null hypothesis that the data is normal. This does NOT prove the data is normal — it just means the data is *consistent* with a normal distribution. The W statistic of 0.987 (close to 1) also suggests approximate normality.18. Which statement best captures the threshold concept of this chapter?
(a) The normal distribution is the only important distribution in statistics (b) All data is normally distributed if you collect enough of it (c) The normal distribution is a useful mathematical model even though no real data is perfectly normal (d) You should only use the normal distribution when the Shapiro-Wilk test gives $p > 0.05$
Answer
**(c) The normal distribution is a useful mathematical model even though no real data is perfectly normal.** This captures George Box's "all models are wrong, but some are useful" principle. The question is never "Is my data normal?" (the answer is always no). The question is "Is my data close enough to normal that the normal model gives useful answers?" This requires judgment, not just a test.19. A Shapiro-Wilk test on a sample of $n = 50{,}000$ gives $p = 0.001$. Which interpretation is most appropriate?
(a) The data is severely non-normal and the normal model should never be used (b) The data has a statistically detectable departure from normality, but it may still be practically close enough to normal for many purposes (c) The test is broken because the sample is too large (d) The p-value should be compared to a stricter threshold like $p < 0.0001$
Answer
**(b) The data has a statistically detectable departure from normality, but it may still be practically close enough to normal for many purposes.** With very large samples, the Shapiro-Wilk test can detect trivially small departures from normality that have no practical impact on your analysis. A small p-value with $n = 50{,}000$ doesn't mean the data is "far" from normal — it means the test has enough power to detect even tiny deviations. Always pair the test with a QQ-plot to assess whether the departure is practically meaningful.20. For a binomial distribution with $n = 100$ and $p = 0.5$, the normal approximation gives $\mu = 50$ and $\sigma = 5$. Using the normal model, approximately what is $P(45 \leq X \leq 55)$?
(a) About 50% (b) About 68% (c) About 95% (d) About 99.7%
Answer
**(b) About 68%.** The range from 45 to 55 is $\mu \pm \sigma = 50 \pm 5$, which is exactly 1 standard deviation on each side of the mean. By the Empirical Rule (or the 68-95-99.7 property of the normal distribution), approximately 68% of the distribution falls in this range.Scoring Guide
| Score | Interpretation |
|---|---|
| 18-20 | Excellent — you've mastered probability distributions and the normal curve |
| 15-17 | Good — solid understanding with a few gaps to review |
| 12-14 | Fair — review Sections 10.3-10.7 and rework the examples |
| Below 12 | Review the chapter carefully, focusing on the worked examples and Python code |