Year 12 Statistics Answers

Name:

Date:

📊 Part 1: Multiple Choice Answers

1. Which measure of centre is most affected by outliers?

Mode

Median

✓

Mean

Range

Explanation: The mean is most affected by outliers because all values are used in its calculation.

2. A correlation coefficient of -0.85 indicates:

Weak positive correlation

✓

Strong negative correlation

No correlation

Perfect positive correlation

Explanation: Values close to -1 indicate strong negative correlation.

3. In a normal distribution, approximately what percentage of data falls within one standard deviation of the mean?

✓

68%

95%

99.7%

50%

Explanation: The empirical rule states that 68% of data falls within 1 standard deviation.

4. Which sampling method would be most appropriate for surveying students about school lunches?

Convenience sampling

✓

Stratified random sampling

Systematic sampling

Cluster sampling

Explanation: Stratified sampling ensures representation from different year levels.

📈 Part 2: Calculation Problems

5. Calculate the mean, median, and mode for the dataset: 12, 15, 18, 15, 22, 19, 15, 20

ACHIEVED LEVEL ANSWER:

Ordered data: 12, 15, 15, 15, 18, 19, 20, 22

Mean = (12+15+15+15+18+19+20+22) ÷ 8 = 136 ÷ 8 = 17

Median = (15+18) ÷ 2 = 16.5

Mode = 15 (appears 3 times)

6. For a dataset with mean = 50 and standard deviation = 8, find the z-score for x = 66

MERIT LEVEL ANSWER:

z = (x - μ) ÷ σ

z = (66 - 50) ÷ 8

z = 16 ÷ 8 = 2

Interpretation: The value 66 is 2 standard deviations above the mean.

7. A study shows that 65% of teenagers own smartphones. In a random sample of 200 teenagers, what is the expected number and standard deviation of smartphone owners?

EXCELLENCE LEVEL ANSWER:

This follows a binomial distribution: n = 200, p = 0.65

Expected value (μ) = np = 200 × 0.65 = 130

Standard deviation (σ) = √(np(1-p)) = √(200 × 0.65 × 0.35)

σ = √(45.5) = 6.75

Interpretation: We expect about 130 smartphone owners, with typical variation of ±6.75.

🎯 Part 3: Statistical Inference

8. A researcher claims the average height of Year 12 students is 170cm. A sample of 25 students has a mean height of 172cm with a standard deviation of 6cm. Test this claim at α = 0.05.

EXCELLENCE LEVEL ANSWER:

Step 1: H₀: μ = 170cm, H₁: μ ≠ 170cm (two-tailed test)

Step 2: Test statistic: t = (x̄ - μ) ÷ (s/√n)

t = (172 - 170) ÷ (6/√25) = 2 ÷ 1.2 = 1.67

Step 3: df = 24, critical value = ±2.064 (α = 0.05)

Step 4: Since |1.67| < 2.064, we fail to reject H₀

Conclusion: There is insufficient evidence to reject the claim that the average height is 170cm.

9. Explain the difference between Type I and Type II errors in hypothesis testing.

MERIT/EXCELLENCE LEVEL ANSWER:

Type I Error: Rejecting the null hypothesis when it is actually true. The probability of this occurring is α (significance level). Also called a "false positive".

Type II Error: Failing to reject the null hypothesis when it is actually false. The probability of this occurring is β. Also called a "false negative".

Example: In medical testing, Type I would be diagnosing a healthy person as sick, while Type II would be failing to diagnose a sick person.

10. A confidence interval for the mean weight of apples is calculated as (145g, 155g). Interpret this result and explain what the confidence level means.

EXCELLENCE LEVEL ANSWER:

Interpretation: We are confident that the true population mean weight of apples lies between 145g and 155g.

Confidence Level: If this procedure were repeated many times with different samples, the stated percentage of intervals would contain the true population mean.

Note: The confidence level refers to the method, not this specific interval. We cannot say there's a certain probability that μ lies in this interval.