A Level H1 Mathematics Statistics Probability Quiz

A-Level Maths H1 Quiz - Statistics Probability

Name: _________________ Class: _________________ Date: _________________

Score: _____ / 30 Duration: 45 minutes

Instructions:

• Answer ALL questions in the spaces provided
• Show all working clearly
• Give answers to appropriate degree of accuracy as specified
• A graphing calculator may be used unless otherwise stated

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Section A: Probability [12 marks]

Question 1 [2 marks] A bag contains 8 red balls and 12 blue balls. Two balls are drawn at random without replacement.

Draw a probability tree diagram to show all possible outcomes for the two draws.

Answer:

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Question 2 [3 marks] In a survey of 200 smartphone users, 60% have anti-virus software installed. A random sample of 15 users is selected.

Find the probability that more than 10 users in the sample have anti-virus software installed.

Answer:

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Question 3 [2 marks] A quality control inspector tests electronic components. The probability that a component is defective is 0.05. Components are tested independently.

Find the probability that in a batch of 20 components, exactly 2 are defective.

Answer:

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Section B: Sampling and Estimation [10 marks]

Question 4 [1 mark] A researcher wants to select a random sample of 50 residents from a town of 2000 residents.

Describe an alternative sampling method that will ensure each resident has an equal chance of being selected.

Answer:

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Question 5 [4 marks] The ages (in months) of 8 toddlers when they first walked are: 12, 14, 11, 15, 13, 16, 12, 14

Calculate the unbiased estimates of the population mean and variance.

Answer: Mean = _________________ months

Variance = _________________ months²

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Question 6 [3 marks] A sample of size n is taken from a population with mean μ = 25 and standard deviation σ = 4.

The sample mean X̄ has a standard deviation of 0.8.

Find the value of n.

Answer:

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Question 7 [2 marks] State the conditions under which the Central Limit Theorem applies to the distribution of sample means.

Answer:

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Section C: Correlation and Regression [8 marks]

Question 8 [2 marks] The table shows data for 6 students relating hours studied (x) and test scores (y):

Hours (x)	2	4	6	8	10	12
Score (y)	45	55	65	75	85	95

Give a sketch of the scatter diagram for this data.

Answer:

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Question 9 [3 marks] For the data in Question 8, find the equation of the least squares regression line of y on x in the form y = mx + c.

Give your values of m and c to 3 significant figures.

Answer: y = _________________ x + _________________

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Question 10 [3 marks] A researcher finds that the correlation coefficient between two variables is r = 0.85.

Comment on the strength and direction of the linear relationship between the variables, and explain what this means in practical terms.

Answer:

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END OF QUIZ

A-Level Maths H1 Quiz - Statistics Probability - ANSWERS

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Section A: Probability [12 marks]

Question 1 [2 marks] Answer:

First draw:     Second draw:
    8/20 ——— Red ——— 7/19 ——— Red
     |               12/19 ——— Blue
     |
    12/20 ——— Blue ——— 8/19 ——— Red
                      11/19 ——— Blue

Marking: 1 mark for correct first stage probabilities, 1 mark for correct second stage conditional probabilities

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Question 2 [3 marks] Let X = number of users with anti-virus software X ~ B(15, 0.6) P(X > 10) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) Using calculator: P(X > 10) = 1 - P(X ≤ 10) = 1 - 0.7827 = 0.217

Marking: 1 mark for identifying binomial distribution, 1 mark for correct setup, 1 mark for correct answer

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Question 3 [2 marks] Let X = number of defective components X ~ B(20, 0.05) P(X = 2) = C(20,2) × (0.05)² × (0.95)¹⁸ = 190 × 0.0025 × 0.3972 = 0.189

Marking: 1 mark for correct binomial setup, 1 mark for correct calculation

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Section B: Sampling and Estimation [10 marks]

Question 4 [1 mark] Answer: Assign each resident a number from 1 to 2000. Use a random number generator to select 50 distinct numbers. Interview the residents corresponding to these numbers.

Marking: 1 mark for describing a valid random sampling method

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Question 5 [4 marks] Answer: Sample mean: x̄ = (12+14+11+15+13+16+12+14)/8 = 107/8 = 13.375 months

Sample variance: s² = Σ(xᵢ - x̄)²/(n-1) = [(12-13.375)² + (14-13.375)² + ... + (14-13.375)²]/7 = [1.891 + 0.391 + 5.641 + 2.641 + 0.141 + 6.891 + 1.891 + 0.391]/7 = 19.875/7 = 2.839 months²

Marking: 2 marks for correct mean, 2 marks for correct unbiased variance (using n-1)

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Question 6 [3 marks] Standard deviation of sample mean = σ/√n 0.8 = 4/√n √n = 4/0.8 = 5 n = 25

Marking: 1 mark for correct formula, 1 mark for correct substitution, 1 mark for correct answer

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Question 7 [2 marks] Answer:

1. The sample size n should be large (typically n > 30)
2. The samples should be drawn independently from the population

Marking: 1 mark for each correct condition

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Section C: Correlation and Regression [8 marks]

Question 8 [2 marks] Answer: Scatter diagram should show:

• x-axis labeled "Hours" (0 to 12)
• y-axis labeled "Score" (40 to 100)
• 6 points plotted: (2,45), (4,55), (6,65), (8,75), (10,85), (12,95)
• Points showing clear positive linear relationship

Marking: 1 mark for correct axes and labels, 1 mark for correctly plotted points

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Question 9 [3 marks] x̄ = 7, ȳ = 70 Sₓₓ = Σ(x-x̄)² = 70 Sₓᵧ = Σ(x-x̄)(y-ȳ) = 350 m = Sₓᵧ/Sₓₓ = 350/70 = 5.00 c = ȳ - mx̄ = 70 - 5×7 = 35.0

Answer: y = 5.00x + 35.0

Marking: 1 mark for calculating means, 1 mark for correct gradient, 1 mark for correct intercept

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Question 10 [3 marks] Answer: The correlation coefficient r = 0.85 indicates a strong positive linear relationship between the variables. This means that as one variable increases, the other tends to increase as well. The relationship is strong because r is close to 1, and about 72% (r² = 0.72) of the variation in one variable can be explained by the linear relationship with the other variable.

Marking: 1 mark for identifying strong positive relationship, 1 mark for explaining direction, 1 mark for practical interpretation

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TOTAL: 30 marks