Hypothesis Testing Practice Questions with Answers

Hypothesis testing is a formal statistical framework used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. By comparing a null hypothesis against an alternative hypothesis, researchers can make objective, data-driven decisions while accounting for the inherent variability in sampling. Whether you are analyzing clinical trials or industrial quality control, mastering statistical hypothesis testing is essential for interpreting scientific data accurately.

Concept Explanation

Hypothesis testing is a systematic procedure for deciding whether the results of a research study support a particular theory which applies to a population. It operates on the principle of falsification, where we start with a Null Hypothesis (H₀), representing the status quo or no effect, and an Alternative Hypothesis (H₁), representing the claim we seek to prove. To reach a conclusion, we calculate a test statistic and compare it to a critical value or use a p-value to determine the probability of observing our results if the null hypothesis were true.

Key Components of Hypothesis Testing

• Significance Level (α): The threshold for rejecting the null hypothesis, commonly set at 0.05 or 5%.

• Type I Error: Rejecting the null hypothesis when it is actually true (a "false positive").

• Type II Error: Failing to reject the null hypothesis when it is actually false (a "false negative").

• P-value: The probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.

The process generally follows five steps: stating the hypotheses, choosing the significance level, collecting data and calculating a test statistic, determining the p-value or critical value, and making a final decision. This rigorous approach prevents researchers from making claims based on random chance. For those looking to sharpen their analytical skills in other scientific areas, practicing with polarity determination questions can help build a strong foundation in data interpretation.

Solved Examples

Following these worked examples will help you understand the mathematical application of hypothesis testing in real-world scenarios.

Example 1: One-Sample Z-Test for Means

A lightbulb manufacturer claims their bulbs last 1,200 hours on average with a standard deviation of 50 hours. A consumer group tests 100 bulbs and finds an average life of 1,185 hours. Test the claim at a 0.05 significance level.

1. State Hypotheses: H₀: μ = 1200; H₁: μ ≠ 1200 (Two-tailed test).

2. Identify Given Values: x̄ = 1185, μ = 1200, σ = 50, n = 100, α = 0.05.

3. Calculate Standard Error: SE = σ / √n = 50 / √100 = 5.

4. Calculate Z-statistic: Z = (1185 - 1200) / 5 = -3.0.

5. Determine Critical Value: For α = 0.05, critical Z-values are ±1.96.

6. Conclusion: Since |-3.0| > 1.96, we reject H₀. There is sufficient evidence that the bulbs do not last 1,200 hours.

Example 2: One-Sample T-Test (Small Sample)

A nutritionist claims a new diet reduces weight by 5 lbs in a month. A sample of 9 people showed a mean loss of 4 lbs with a sample standard deviation of 1.2 lbs. Test at α = 0.01.

1. State Hypotheses: H₀: μ = 5; H₁: μ < 5 (Left-tailed test).

2. Identify Values: x̄ = 4, μ = 5, s = 1.2, n = 9, df = 8.

3. Calculate T-statistic: T = (4 - 5) / (1.2 / √9) = -1 / 0.4 = -2.5.

4. Find Critical Value: Looking at a T-distribution table for df = 8 and α = 0.01, the critical value is -2.896.

5. Conclusion: Since -2.5 is not less than -2.896, we fail to reject H₀. There is not enough evidence to say the weight loss is less than 5 lbs.

Example 3: Proportion Testing

A company claims 90% of customers are satisfied. A survey of 200 customers finds 170 are satisfied. Test at α = 0.05.

1. State Hypotheses: H₀: p = 0.90; H₁: p < 0.90.

2. Identify Values: p₀ = 0.90, p̂ = 170/200 = 0.85, n = 200.

3. Calculate Standard Error: SE = √[(0.9 * 0.1) / 200] = √0.00045 ≈ 0.0212.

4. Calculate Z-statistic: Z = (0.85 - 0.90) / 0.0212 = -2.36.

5. Conclusion: The critical Z-value for a one-tailed test at α = 0.05 is -1.645. Since -2.36 < -1.645, we reject H₀. Satisfaction is significantly lower than claimed.

Practice Questions

1. A coffee shop claims their medium cup contains 12 ounces of coffee. A researcher measures 36 cups and finds a mean of 11.8 ounces with a population standard deviation of 0.6 ounces. Test the claim at α = 0.05.

2. A medication is known to be effective for 70% of patients. A new study of 100 patients shows 80 patients improved. Test if the new medication is more effective at α = 0.01.

3. A factory produces steel rods with a mean diameter of 2.5 cm. A sample of 16 rods has a mean of 2.55 cm and a sample standard deviation of 0.08 cm. Test if the mean diameter has changed at α = 0.05.

4. A researcher believes that the average commute time in a city is greater than 30 minutes. A sample of 50 commuters shows a mean of 33 minutes with a population standard deviation of 8 minutes. Test at α = 0.05.

5. A seed company claims that 95% of their seeds germinate. Out of 400 seeds, 372 germinate. Is the germination rate significantly lower than claimed? (α = 0.05).

6. In a comparison of two teaching methods, Group A (n=30) had a mean score of 82 and Group B (n=30) had a mean score of 78. Assume a known population standard deviation of 5 for both. Test if Method A is better at α = 0.05.

7. A car's braking distance at 60 mph is claimed to be 120 feet. A test of 25 cars shows a mean distance of 124 feet with a sample standard deviation of 10 feet. Test at α = 0.05.

8. A psychologist wants to know if a new therapy reduces anxiety scores. The mean score before therapy was 50. After therapy, 16 patients had a mean score of 46 with a sample standard deviation of 8. Test at α = 0.05.

9. A soft drink machine is set to discharge 8.0 ounces per cup. A sample of 10 cups yields a mean of 7.8 ounces and a sample standard deviation of 0.15 ounces. Test if the machine is underfilling at α = 0.01.

10. A poll suggests that 52% of voters support a new law. A random sample of 500 voters shows 240 support it. Test if the support is actually less than 52% at α = 0.05.

Answers & Explanations

1. Reject H₀. Z = (11.8 - 12) / (0.6 / √36) = -0.2 / 0.1 = -2.0. Critical Z for α=0.05 (two-tailed) is ±1.96. Since |-2.0| > 1.96, the claim is rejected.

2. Reject H₀. SE = √[(0.7 * 0.3) / 100] = 0.0458. Z = (0.8 - 0.7) / 0.0458 = 2.18. Critical Z for α=0.01 (one-tailed) is 2.33. Wait, 2.18 < 2.33, so we Fail to Reject H₀. (Correction: Evidence is not strong enough at the 1% level).

3. Fail to Reject H₀. T = (2.55 - 2.5) / (0.08 / √16) = 0.05 / 0.02 = 2.5. Critical T for df=15, α=0.05 (two-tailed) is 2.131. Actually, since 2.5 > 2.131, we Reject H₀.

4. Reject H₀. Z = (33 - 30) / (8 / √50) = 3 / 1.13 = 2.65. Critical Z for α=0.05 (right-tailed) is 1.645. Since 2.65 > 1.645, the average commute is significantly higher.

5. Reject H₀. p̂ = 372/400 = 0.93. SE = √[(0.95 * 0.05) / 400] = 0.0109. Z = (0.93 - 0.95) / 0.0109 = -1.83. Critical Z for α=0.05 (left-tailed) is -1.645. Since -1.83 < -1.645, it is significantly lower.

6. Reject H₀. SE_diff = √[(5²/30) + (5²/30)] = 1.29. Z = (82 - 78) / 1.29 = 3.10. Critical Z is 1.645. Method A is significantly better.

7. Fail to Reject H₀. T = (124 - 120) / (10 / √25) = 4 / 2 = 2.0. Critical T for df=24, α=0.05 (two-tailed) is 2.064. Since 2.0 < 2.064, we do not have enough evidence.

8. Reject H₀. T = (46 - 50) / (8 / √16) = -4 / 2 = -2.0. Critical T for df=15, α=0.05 (one-tailed) is -1.753. Since -2.0 < -1.753, the therapy is effective.

9. Reject H₀. T = (7.8 - 8.0) / (0.15 / √10) = -0.2 / 0.0474 = -4.22. Critical T for df=9, α=0.01 (one-tailed) is -2.821. The machine is underfilling.

10. Reject H₀. p̂ = 240/500 = 0.48. SE = √[(0.52 * 0.48) / 500] = 0.0223. Z = (0.48 - 0.52) / 0.0223 = -1.79. Critical Z for α=0.05 (left-tailed) is -1.645. Support is significantly less than 52%.

To deepen your understanding of complex analysis, you might also find reaction mechanism practice questions useful for learning how to step through logical proofs.

Quick Quiz

Frequently Asked Questions

What is the difference between a one-tailed and two-tailed test?

A one-tailed test looks for a change in a specific direction (e.g., "greater than"), while a two-tailed test looks for any change regardless of direction (e.g., "different from"). Choosing between them depends on whether your research hypothesis specifies a direction.

Why is 0.05 commonly used as the significance level?

The 0.05 level was popularized by Ronald Fisher as a convenient standard for scientific research, representing a 1 in 20 chance of being wrong. However, it is an arbitrary threshold and can be adjusted depending on the consequences of making a Type I or Type II error.

What happens to the p-value as the sample size increases?

Generally, as the sample size increases, the standard error decreases, which typically leads to a larger test statistic and a smaller p-value if a true effect exists. Larger samples provide more power to detect even small differences in the population.

Can you ever "prove" the null hypothesis is true?

No, hypothesis testing never proves the null hypothesis; we only "fail to reject" it. This means the evidence is insufficient to support the alternative, not that the null is definitively true.

What is the power of a statistical test?

The power of a test (1 - β) is the probability that the test correctly rejects a false null hypothesis. High power reduces the risk of a Type II error, ensuring that real effects are actually detected by the study.

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