Use this diagnostic to find which Unit 4 topics to revisit: one short question per topic, two for Topics 4.2, 4.5 and 4.10.
These are original Marlbridge practice questions, not past exam questions, with fictional data. The diagnostic is not calibrated and gives no predicted score.
How to take it. Allow 30 minutes; answer everything before opening any answer. Graphing calculator or t-table allowed. Two-sample procedures are not pooled. Otherwise assume samples are random and the 10% condition holds.
Question 1 (multiple choice · 4.1)
Songs on a fictional streaming platform have mean length 214 seconds and standard deviation 48 seconds. An analyst selects a random sample of 36 of its millions of songs. Which describes the sampling distribution of x̄?
- (A) Mean 214 s, standard deviation 48 s
- (B) Mean 214 s, standard deviation 8 s
- (C) Mean 5.94 s, standard deviation 8 s
- (D) Mean 214 s, standard deviation 1.33 s
Answer and explanation
Answer: (B). μx̄ = μ = 214 s and σx̄ = σ/√n = 48/√36 = 8 s (random sample; 36 is far below 10% of millions).
- (A) gives the spread of single songs.
- (C) divides the mean by n; the centre does not change.
- (D) divides σ by n instead of √n.
If you missed this: Topic 4.1 study guide.
Question 2 (multiple choice · 4.2)
A random sample of 16 of a fictional pharmacy’s 4,000 monthly prescriptions had mean waiting time 48.3 minutes, s = 6.0 minutes, with no strong skewness or outliers. Which is the 95% confidence interval for the mean waiting time?
- (A) (45.36, 51.24) minutes
- (B) (45.10, 51.50) minutes
- (C) (35.51, 61.09) minutes
- (D) (47.50, 49.10) minutes
Answer and explanation
Answer: (B). σ is unknown, so use t with df = 15: t* = 2.131. MOE = 2.131 × 6.0/√16 = 3.197 minutes.
- (A) uses z* = 1.96.
- (C) multiplies t* by s, not by s/√n.
- (D) divides s by n instead of √n.
If you missed this: Topic 4.2 study guide.
Question 3 (short answer · 4.2)
A fictional supermarket sells 5,000 products in store and online. A shopper picks 10 at random and records both prices. The differences d = in-store − online, in cents, are:
12, 5, −3, 8, 15, 0, 6, 9, −2, 10
(a) Why are these matched pairs? Define the parameter. (b) Check the conditions. (c) Construct a 95% confidence interval.
Answer and explanation
(a) Each product gives two linked prices: one sample of 10 differences. μd = the true mean difference (in-store − online) in price, in cents, for all 5,000 products.
(b) Random sample of products ✓. 10%: 10 ≤ 500 ✓. n = 10 < 30: Q1 = 0, Q3 = 10, fences −15 and 25; no outliers or strong skew ✓.
(c) x̄d = 6.0, sd = 6.0369 cents. df = 9, t* = 2.262. MOE = 2.262 × 6.0369/√10 = 4.3186. Interval: (1.68, 10.32) cents.
If you missed this: Topic 4.2 study guide, “Worked example 2”.
Question 4 (multiple choice · 4.3)
A fictional soup’s label says 600 mg of sodium per can. From a random sample of 30 cans, a 99% t-interval for the true mean sodium content is (612, 655) mg. Is there convincing evidence that the true mean differs from the label?
- (A) Yes: 600 mg is not in the interval, so it is not a plausible mean.
- (B) No: some cans in the sample may contain less than 600 mg.
- (C) Yes: 99% of cans contain between 612 and 655 mg.
- (D) No: only a significance test can judge a claim.
Answer and explanation
Answer: (A). Every plausible value of μ is above 600 mg.
- (B) confuses individual cans with the mean.
- (C) treats the interval as a range for individual cans.
- (D) is wrong: an interval that excludes a claimed value is evidence against it.
If you missed this: Topic 4.3 study guide.
Question 5 (multiple choice · 4.4)
A fictional sleep app says its bedtime routine reduces the time users take to fall asleep. Twenty-five randomly chosen users record their mean time to fall asleep for a week before and a week after starting it. With d = before − after, which hypotheses fit the claim?
- (A) H₀: μd = 0, Hₐ: μd < 0
- (B) H₀: x̄d = 0, Hₐ: x̄d > 0
- (C) H₀: μd = 0, Hₐ: μd > 0
- (D) H₀: μd = 0, Hₐ: μd ≠ 0
Answer and explanation
Answer: (C). If the routine helps, before − after is positive. μd is the true mean difference (before − after) for all the app’s users.
- (A) reverses the direction for this order of subtraction.
- (B) uses the sample statistic x̄d; hypotheses are about parameters.
- (D) is two-sided; the claim names a direction.
If you missed this: Topic 4.4 study guide.
Question 6 (multiple choice · 4.5)
A one-sample t-test with n = 14 and Hₐ: μ > μ₀ gives t = 2.05. Find the p-value.
- (A) 0.0202
- (B) 0.0305
- (C) 0.0611
- (D) 0.9695
Answer and explanation
Answer: (B). df = 13; p-value = P(t ≥ 2.05) = 0.0305, the upper tail only.
- (A) uses the standard normal instead of t with 13 df.
- (C) doubles for a two-sided test.
- (D) is the area in the wrong tail.
If you missed this: Topic 4.5 study guide.
Question 7 (short answer · 4.5)
A fictional council says its recycling trucks take 95 minutes, on average, to complete a collection route. Residents suspect it takes longer. A random sample of 20 of last year’s 1,500 route runs gives x̄ = 99.2 minutes and s = 8.1 minutes; a dotplot shows no strong skewness or outliers. The hypotheses are H₀: μ = 95, Hₐ: μ > 95.
(a) Calculate the test statistic and p-value. (b) Interpret the p-value in context. (c) State a conclusion at α = 0.05.
Answer and explanation
(a) SE = 8.1/√20 = 1.8112. t = (99.2 − 95)/1.8112 = 2.32, df = 19. p-value = P(t ≥ 2.32) = 0.0158.
(b) Assuming the true mean route time is 95 minutes, there is a 0.0158 probability of a sample mean of 99.2 minutes or more in a random sample of 20 runs.
(c) Because 0.0158 ≤ 0.05, reject H₀. There is convincing evidence that the true mean time of all last year’s route runs exceeds 95 minutes.
If you missed this: Topic 4.5 study guide.
Question 8 (multiple choice · 4.6)
Waiting times at fictional theme-park Ride A are normal with mean 32 and standard deviation 9 minutes; at Ride B, normal with mean 27 and standard deviation 6 minutes. Independent random samples of 9 waits at A and 16 at B are taken. What is the probability that the B sample mean is greater than the A sample mean?
- (A) 0.0680
- (B) 0.1333
- (C) 0.3220
- (D) 0.9320
Answer and explanation
Answer: (A). x̄A − x̄B is normal (both populations are normal) with mean 5 and standard deviation √(81/9 + 36/16) = 3.3541 minutes. P(x̄A − x̄B < 0) = P(z < −1.4907) = 0.0680.
- (B) adds standard deviations (3 + 1.5), not variances.
- (C) forgets to divide each variance by its n.
- (D) is the wrong tail.
If you missed this: Topic 4.6 study guide.
Question 9 (multiple choice · 4.7)
A two-sample t-interval uses samples of 14 and 20. Which df could technology report?
- (A) 12.0
- (B) 27.6
- (C) 33.0
- (D) 34.0
Answer and explanation
Answer: (B). The df lie between the smaller of n₁ − 1 and n₂ − 1 (13) and n₁ + n₂ − 2 (32).
- (A) is below 13.
- (C) is above 32.
- (D) is n₁ + n₂.
If you missed this: Topic 4.7 study guide.
Question 10 (multiple choice · 4.8)
Sixty volunteers were randomly assigned, 30 each, to read a passage in a serif or a sans-serif typeface. A 95% interval for μ_serif − μ_sans, the difference in mean reading time, is (−7.4, −1.2) seconds. A designer claims sans-serif text is read faster. Which is correct?
- (A) Supported, because the interval is entirely negative.
- (B) Not supported: the serif mean time seems shorter, by 1.2 to 7.4 seconds, for volunteers like these.
- (C) No conclusion is possible, because the interval does not contain 0.
- (D) Supported: 95% of volunteers read sans-serif 1.2 to 7.4 seconds faster.
Answer and explanation
Answer: (B). Negative values mean serif times are smaller; 0 is not plausible, so the evidence points the other way.
- (A) ignores the order of subtraction.
- (C) is backwards: excluding 0 gives evidence.
- (D) describes individuals and reverses the direction.
If you missed this: Topic 4.8 study guide.
Question 11 (multiple choice · 4.9)
A sports scientist randomly assigns 24 volunteer runners, 12 to drink X and 12 to drink Y, and records how far each runs in 30 minutes. She suspects drink X leads to a longer mean distance. Which set-up is correct?
- (A) Two-sample t-test; μ_X and μ_Y are the true mean distances for runners like these with each drink; Hₐ: μ_X − μ_Y > 0
- (B) As (A), but μ_X and μ_Y are the means of the 12 runners in each group
- (C) As (A), but Hₐ: μ_X − μ_Y ≠ 0
- (D) One-sample t-test for μd, the mean of the 12 differences X − Y
Answer and explanation
Answer: (A). Two independent groups; true means as parameters; a one-sided Hₐ.
- (B) defines sample means, not parameters.
- (C) is two-sided.
- (D) treats unrelated runners as pairs.
If you missed this: Topic 4.9 study guide.
Question 12 (multiple choice · 4.10)
A lab suspects fictional Brand 1 smoke alarms sound faster than Brand 2. Random samples of 20 of each give t = −2.48 with df = 37.6 for H₀: μ₁ − μ₂ = 0, Hₐ: μ₁ − μ₂ < 0. Which is correct at α = 0.01?
- (A) p ≈ 0.0089; reject H₀: convincing evidence Brand 1’s true mean time is shorter
- (B) p ≈ 0.0177; fail to reject H₀
- (C) p ≈ 0.9911; fail to reject H₀
- (D) p ≈ 0.0089; reject H₀: this proves Brand 1 is faster
Answer and explanation
Answer: (A). P(t ≤ −2.48) with df = 37.6 is 0.0089 ≤ 0.01.
- (B) doubles for a two-sided test.
- (C) is the wrong tail.
- (D) uses definite language; tests give evidence, not proof.
If you missed this: Topic 4.10 study guide.
Question 13 (short answer · 4.10)
A fictional streaming service randomly assigned 80 volunteers, 40 each, to a new or old home page and timed how long each took to start a programme. New: x̄ = 3.4, s = 1.6 minutes. Old: x̄ = 4.3, s = 1.9 minutes. H₀: μ_new − μ_old = 0, Hₐ: μ_new − μ_old < 0; the conditions are met.
(a) Find t, df and the p-value. (b) State a conclusion at α = 0.05. (c) What does the design allow?
Answer and explanation
(a) SE = √(1.6²/40 + 1.9²/40) = 0.3927 minutes. t = (3.4 − 4.3)/0.3927 = −2.29. Technology: df = 75.80, p-value = P(t ≤ −2.29) = 0.0124.
(b) Because 0.0124 ≤ 0.05, reject H₀. There is convincing evidence that the true mean time to start a programme is shorter with the new page.
(c) Pages were randomly assigned, so the new page caused the shorter mean time, for volunteers like these.
If you missed this: Topic 4.10 study guide.
Your next step
| Topic | Question(s) | If you missed it, read |
|---|---|---|
| 4.1 | 1 | Sampling distribution of x̄ |
| 4.2 | 2, 3 | One-sample t-interval |
| 4.3 | 4 | Claims from an interval |
| 4.4 | 5 | Setting up a t-test |
| 4.5 | 6, 7 | Carrying out a t-test |
| 4.6 | 8 | Sampling distribution of x̄₁ − x̄₂ |
| 4.7 | 9 | Two-sample t-interval |
| 4.8 | 10 | Claims about two means |
| 4.9 | 11 | Two-sample test set-up |
| 4.10 | 12, 13 | Two-sample test |
How to use your result
- Mark each question right, partly right or wrong. A number with no conclusion in context is only partly right.
- Fix weak topics in order: tests (4.4, 4.5, 4.9, 4.10) reuse the standard errors and conditions of 4.1, 4.2, 4.6, 4.7.
- Read the study guide, then do its practice set.
- Then try the Unit 4 mixed review.