Master the logic of null & alternative hypotheses, margin of error, and drawing conclusions from data.
20
Questions
4
Difficulty Levels
HSS.IC
Standards
Quick Reference
Core Concepts & Memory Points
Study these before starting — they are your cheat sheet for every question below.
🔑 Key Terms
Null vs. Alternative
H₀ = "nothing changed" / claim is true Hₐ = "something IS different" / claim is false
H₀ always uses =
Hₐ uses ≠ > <
📐 Formula
Margin of Error
MOE tells you how far the sample mean can stray from the true mean.
MOE = 2 × (σ / √n)
σ = std dev n = sample size
✅ Decision Rule
Accept or Reject H₀?
Calculate the confidence interval: x̄ ± MOE
If the claim falls inside → cannot reject H₀
If the claim falls outside → Reject H₀
⚠️ Common Trap
"Fail to Reject" ≠ "Accept"
We never say "accept H₀." We only say:
• Reject H₀
• Fail to reject H₀
Not enough evidence ≠ proof it's true.
🎯 Instant Recall
4 Steps to Evaluate a Claim
① Write H₀ and Hₐ
② Calculate MOE = 2σ/√n
③ Build interval: x̄ ± MOE
④ Is the claim inside the interval?
📝 Bigger n
Sample Size Effect
Larger sample → smaller MOE → narrower interval → easier to reject a false claim.
n↑ ⟹ √n↑ ⟹ MOE↓
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Level 1 · EasyVocabulary & Basic Concepts
1
Concept Check
In statistics, a hypothesis is best described as:
2
Concept Check
Which symbol is always used in the null hypothesis (H₀)?
3
True / False Concept
A researcher fails to reject H₀. Which statement is correct?
4
Formula Identification
The Margin of Error (MOE) formula uses which two values?
5
Calculation · Basic
A study has σ = 0.20 and n = 100. What is the Margin of Error? (Use: MOE = 2σ/√n)
Level 2 · MediumWriting Hypotheses & Intervals
6
Writing Hypotheses
A car company claims its new model averages 35 mpg. You want to test whether the true mean is different from 35. Which pair of hypotheses is correct?
7
Writing Hypotheses · One-Tailed
A principal believes the mean student GPA is at most 3.2. A researcher suspects it is actually higher. Write the correct hypotheses.
8
Interval Construction
A sample has x̄ = $1.64 and MOE = $0.023. What is the confidence interval for the true mean?
9
Decision Rule
A study yields the interval (48.2, 51.8). The company claims the mean is 53. What is the correct conclusion?
10
Effect of Sample Size
Two studies test the same claim. Study A uses n = 50; Study B uses n = 200. Which statement is true about their margins of error?
Level 3 · HardWord Problems — Test Difficulty
11
Word Problem · Full Process
Mac&Cheese Price Study
Best Bet Mac&Cheese reports its average price is $1.45. A marketing company surveys 100 retail prices and finds x̄ = $1.65, with σ = $0.20. Does this study provide strong evidence that the claim is false?
What is the confidence interval and the correct conclusion?
12
Word Problem · Revised Claim
Best Bet Revises Their Claim
Best Bet now claims the average price is $1.60. A second study samples 300 prices and finds x̄ = $1.64, σ = $0.20. Should we reject this revised claim?
13
Word Problem · Golf Driver
Golf Club Study — Identify the Error
A golf club company claims its driver increases distance. A study finds the control group averages 200 ft, the experimental group averages 230 ft. Regina writes: H₀: μ ≥ 230 and Hₐ: μ > 230. What mistake did she make?
14
Word Problem · Fuel Additive
Fuel Additive Test
A car normally gets 34.6 mpg. With a new fuel additive it gets 35.3 mpg. You want to determine whether the additive genuinely increases mileage.
Which pair of hypotheses is correct?
15
Word Problem · Red Light Drivers
A traffic study investigates whether the proportion of drivers who run red lights is greater than 0.5. Let p = the proportion of drivers who admit to running red lights. Which hypotheses are correct?
Level 4 · ExpertMulti-Step & Tricky Word Problems
16
Expert · Multi-Step Calculation
Coffee Chain Study
A coffee chain claims the average wait time is 4 minutes. A consumer group samples 400 visits and finds x̄ = 4.5 minutes, σ = 1.2 minutes. Does this disprove the claim?
Calculate MOE, build the interval, then select the correct conclusion.
17
Expert · Tricky Wording
A study concludes: "We reject the null hypothesis." What does this mean about the alternative hypothesis?
18
Expert · Simulation Logic
A simulation test involves pooling all data values, randomly splitting them into two new groups, and computing the difference in means — repeated many times. What is the purpose of this process?
19
Expert · IQ Score Standard Deviation
A researcher claims the standard deviation of IQ scores of actors is equal to 15. A critic suspects it is not equal to 15. Let σ = standard deviation of IQ scores of actors. Which hypotheses are correct?
20
Expert · Basketball Height Claim
NBA Height Study
A sports analyst believes the mean height of professional basketball players is at most 7 feet. A sample of 64 players finds x̄ = 6.8 ft, σ = 0.4 ft. The analyst writes H₀: M = 7 and Hₐ: M < 7.
Build the confidence interval and determine whether the analyst's H₀ should be rejected.