1.
Explain the difference between a population and a sample, and between a parameter and a statistic.
[3 marks]
- Population: whole group
- Sample: subset
- Parameter from population, statistic from sample
JD SCIENCE
Statistics
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Describe simple random sampling and one advantage and one disadvantage.
[3 marks]
- Every sample of size n equally likely
- Unbiased
- Needs a sampling frame / can be impractical
3.
Describe stratified sampling and when it is better than simple random sampling.
[3 marks]
- Sample in proportion to strata
- More representative when groups differ
4.
Describe opportunity/convenience sampling and why it can be biased.
[2 marks]
- Use who is available
- Not every person has an equal chance
5.
A data set has values 2, 5, 7, 10, 16. Find the mean and the standard deviation (divisor n or n−1 as used on your board).
[4 marks]
- Show the sum of squared deviations
- Final answer: Mean 8; variance = 22.4 (n) or 28 (n−1)
6.
Coded data: y = (x − 20)/2. The mean of y is 3 and the sd of y is 1.4. Find the mean and sd of x.
[3 marks]
- Reverse the coding
- Final answer: Mean x = 26; sd x = 2.8
7.
Interpret a positive skew and a negative skew. Which average is pulled the most?
[3 marks]
- Positive: tail to the right, mean > median
- Mean is pulled toward the tail
8.
Two events A and B: P(A)=0.4, P(B)=0.5, P(A∩B)=0.2. Find P(A∪B) and P(A|B).
[3 marks]
- Addition rule; conditional
- Final answer: 0.7 ; 0.4
9.
State the conditions for a binomial distribution B(n, p).
[3 marks]
- Fixed n
- Two outcomes
- Independent trials
- Constant p
10.
X ~ B(10, 0.3). Find P(X = 2). Leave your answer in a calculable form and evaluate to 3 s.f.
[3 marks]
- Final answer: C(10,2)(0.3)²(0.7)⁸ ≈ 0.233
11.
X ~ B(10, 0.3). Find E(X) and Var(X).
[2 marks]
- np, np(1−p)
- Final answer: 3 and 2.1
12.
X ~ B(20, 0.4). Use your calculator to find P(X ≤ 5).
[2 marks]
- Binomial cdf
- Final answer: 0.1256 (3 s.f., calculator)
13.
When can a normal distribution be used as a model? Mention symmetry and the mean/median.
[2 marks]
- Data roughly symmetric / bell-shaped
- Mean ≈ median
14.
X ~ N(50, 16). Find P(X < 46). Standardise and use Φ.
[3 marks]
- Final answer: z = (46−50)/4 = −1; Φ(−1) = 1 − 0.8413 = 0.1587
15.
X ~ N(100, 25). Find the value a such that P(X < a) = 0.975.
[3 marks]
- Inverse normal
- Final answer: a = 100 + 1.96×5 = 109.8
16.
State the mean and variance of the standard normal Z.
[2 marks]
- 0 and 1
17.
A hypothesis test: H₀: p = 0.5, H₁: p > 0.5, n = 20, observed 14 successes, 5% level. Carry out the test using a binomial model.
[5 marks]
- State conclusion in context
- Final answer: P(X≥14 | B(20,0.5)) = 0.0577 > 0.05, not significant, do not reject H₀
18.
Explain what a Type I error and a Type II error are.
[3 marks]
- Type I: reject H₀ when true
- Type II: fail to reject H₀ when false
19.
Why do we use a continuity correction when approximating a discrete distribution by a normal?
[2 marks]
- Discrete values occupy a unit width
- P(X ≤ 10) ≈ P(Y < 10.5)
20.
PMCC: a scatter graph of revision hours and mark has r = 0.82. Interpret r.
[2 marks]
- Final answer: Strong positive linear correlation
21.
The least-squares regression line is y = 12 + 4.5x. Interpret the gradient and the intercept in context if x is hours and y is mark.
[3 marks]
- Gradient: extra 4.5 marks per hour
- Intercept: predicted mark with 0 hours — may not be meaningful
22.
Why should you not use a regression line to predict far outside the data range?
[2 marks]
- Extrapolation
- The linear model may not hold
23.
A discrete random variable X has P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3. Find E(X) and Var(X).
[4 marks]
- Final answer: E(X)=2.1; E(X²)=4.9; Var=0.49
24.
For that X, find P(X ≥ 2).
[1 mark]
- Final answer: 0.8
25.
Describe a histogram versus a box plot: what can you see on each?
[3 marks]
- Histogram: shape, modality, skew
- Box plot: median, IQR, outliers, easy comparison
26.
Large data set style: a sample of 40 daily maxima has mean 18.2°C. Explain why this mean might not estimate the yearly mean temperature well.
[2 marks]
- Final answer: Only maxima not typical days,Seasonal bias if the 40 days are not spread through the year
27.
Write H₀ and H₁ for a two-tailed test that a coin is fair, and state the meaning of the significance level.
[3 marks]
- H₀: p=0.5, H₁: p≠0.5
- Significance level is P(Type I error)
28.
Y ~ N(0, 1). Find P(−1.96 < Y < 1.96).
[2 marks]
- Standard result
- Final answer: 0.95
29.
On OCR Statistics, you must conclude in context. Write a model concluding sentence for a non-significant result about a plant fertiliser.
[2 marks]
- There is insufficient evidence at the 5% level that the fertiliser increases mean growth
30.
The IQR is 12 and the range is 40. Comment on the presence of possible outliers compared with a data set with IQR 12 and range 16.
[2 marks]
- Final answer: Much larger range suggests outliers or long tails
31.
Explain mutually exclusive vs independent, with a probability condition for each.
[3 marks]
- ME: P(A∩B)=0
- Independent: P(A∩B)=P(A)P(B)
32.
A Venn diagram: 50 students, 30 do maths, 22 do physics, 12 do both. A student is chosen at random. Find P(physics | maths).
[3 marks]
- Final answer: 12/30 = 0.4