Statistics & Probability - Practice Set 1 | Math AI

Q1

The scores of 9 students: 45, 62, 58, 71, 45, 88, 53, 67, 45. Find:

(a) mean,

(b) median,

(c) mode.

Answer

(a) 59.3 (b) 58 (c) 45

Q2

Data: 12, 18, 21, 23, 25, 26, 27, 28, 30, 31, 33, 35, 38, 42. Find Q1, Q3, IQR, and any outliers.

Answer

Q1=22, Q3=33.5, IQR=11.5, no outliers

Q3

A bag has 4 red, 3 blue, 5 green balls. One ball is picked at random. Find:

(a) P(red)

(b) P(not blue)

(c) P(red or green).

Answer

(a) 1/3 (b) 3/4 (c) 3/4

Q4

The correlation coefficient r = −0.92.

(a) Describe the correlation.

(b) What does r² tell you?

(c) Can you conclude that x causes y to decrease?

Answer

(a) Strong negative linear (b) r²≈0.846, 84.6% explained (c) No; correlation ≠ causation

Q5

Describe the difference between:

(a) random sampling

(b) stratified sampling

(c) systematic sampling.

Answer

(a) Every member equally likely (b) Population divided into groups, sample proportional from each (c) Every nth member selected

Q6

X ~ B(10, 0.3). Find:

(a) P(X=3),

(b) P(X≤2),

(c) E(X) and Var(X).

Answer

(a) 0.267 (b) 0.383 (c) E(X)=3, Var=2.1

Q7

A frequency table: marks 10-20 (freq 4), 20-30 (freq 7), 30-40 (freq 12), 40-50 (freq 9), 50-60 (freq 3).

(a) Calculate the mean.

(b) Find the modal class.

(c) Estimate the median.

(d) Draw a cumulative frequency curve description and estimate Q1, Q3.

Answer

(a) 35 (b) 30-40 (c) ≈35.4 (d) Q1≈31.8, Q3≈43.6

Q8

Study time (x hrs/week) vs exam score (y): data pairs (2,45),(5,58),(7,64),(8,69),(10,75),(12,80),(14,88),(15,91).

(a) Find r using GDC.

(b) Find the regression line y=ax+b.

(c) Predict score for 9 hrs/week.

(d) Comment on reliability of prediction for 20 hrs/week.

Answer

(a) r≈0.995 (b) y=3.24x+38.1 (c) ≈67.3 (d) Extrapolation — unreliable

Q9

Events A and B: P(A)=0.4, P(B)=0.5, P(A∩B)=0.25

(a) Are A and B independent?

(b) Find P(A∪B).

(c) Find P(A|B).

(d) Draw a Venn diagram with all regions labelled.

Answer

(a) No (b) 0.65 (c) 0.5 (d) Venn: A∩B=0.25, A only=0.15, B only=0.25, neither=0.35

Q10

A tree diagram: a box has 5 red and 3 blue balls. Two balls drawn without replacement.

(a) Draw the tree diagram.

(b) Find P(both red).

(c) Find P(exactly one of each colour).

(d) Given the second is red, find P(first was also red).

Answer

(a) Tree diagram (b) 5/14 (c) 15/28 (d) 4/7

Q11

X ~ Po(3.5). Find

(a) P(X=4)

(b) P(X≤3)

(c) P(X≥5)

(d) Find E(X) and Var(X).

Answer

(a) 0.189 (b) 0.537 (c) 0.275 (d) E=Var=3.5

Q12

A teacher claims the mean test score is 70. A sample of 25 students gives x̄=66, s=12.

(a) State H₀ and H₁ for a two-tailed test

(b) Perform a t-test at 5% significance level

(c) State your conclusion

(d) Find the 95% confidence interval for the population mean.

Answer

(a) H₀:μ=70, H₁:μ≠70 (b) t=−1.667, not significant (c) Fail to reject H₀ (d) (61.1, 70.9)

Q13

X~N(μ,σ²). P(X<40)=0.2 and P(X<60)=0.85. Find μ and σ.

Answer

σ≈10.65, μ≈49

Q14

Transition matrix: T=[[0.7,0.3],[0.4,0.6]]. Initial state s₀=[0.6,0.4]ᵀ

(a) Find s₁=T·s₀ and s₂=T²·s₀

(b) Find the steady state distribution

(c) Find eigenvalues of T

(d) How many steps until s is within 0.01 of steady state?

Answer

(a) s₁=[0.54,0.46], s₂=[0.516,0.484] (b) [3/7, 4/7] (c) λ=1, λ=0.3 (d) n=5 steps

Q15

X and Y are independent: X~N(10,4), Y~N(6,9). Let W=2X−Y+3

(a) Find E(W)

(b) Find Var(W)

(c) Find P(W>20)

(d) Find P(X+Y<20)

Answer

(a) 17 (b) 25 (c) 0.274 (d) 0.866

Q16

Chi-squared test: a die is rolled 120 times. Results: 1(18),2(22),3(25),4(17),5(16),6(22). At 5% significance level, test if the die is fair.

Answer

χ²=3.1 < 11.07; Fail to reject H₀; die appears fair at 5% level

Q17

The normal distribution graph shows X~N(68,64)

$Stats and probability1 study illustration$

(a) Find P(X > 72) and P(X < 60) from the shaded regions.

(b) Find the value c such that P(|X−68| < c)=0.90.

(c) A student is told they are in the top 15%. Find their minimum score

(d) 5 students are randomly selected. Find P(exactly 3 score above 72).

Answer

(a) 0.309; 0.159 (b) c≈13.2 (c) ≥76.3 (d) ≈0.141

Q18

The box and whisker plot shows data with a clear outlier

$Stats and probability2 study illustration$

(a) Calculate Q1, Q3, IQR, and identify any outliers using the 1.5×IQR rule

(b) Compare mean vs median as measures of centre

(c) Data values: 12,18,21,23,25,26,27,28,30,31,33,35,38,42,67. Find the standard deviation using GDC

(d) Remove the outlier and recalculate mean, median, and SD. Comment on the effect.

Answer

(a) Q1=22, Q3=33.5, IQR=11.5, outlier=67 (b) Median better (c) σ≈12.6 (d) Mean drops to 27.4, SD to 7.1

Q19

Scatter plot: study hours vs exam score, n=12. The regression line ŷ=3.24x+38.1 is shown with r=0.994

$Stats and probability3 study illustration$

(a) Interpret the gradient 3.24 and y-intercept 38.1 in context

(b) Find the score for 11 hours. Is this interpolation or extrapolation?

(c) A student scored 82 and studied 13 hours. Find their residual and interpret

(d) [HL] Test H₀: ρ=0 vs H₁: ρ≠0 at 1% significance level given t=r√(n-2)/√(1-r²).

Answer

(a) +3.24 marks/hour; 38.1 baseline (b) 73.7; interpolation (c) residual=+1.8 (d) Reject H₀; significant

Q20

Binomial distribution X~B(12,0.35). The bar chart highlights P(X≥6)

$Stats and probability4 study illustration$

(a) Find P(X=6) and P(X≥6)

(b) Find the most likely value (mode)

(c) Find E(X), Var(X), SD(X)

(d) Y is defined as the number of trials until the first success (Geometric). If P(success)=0.35, find P(Y=5) and E(Y).

Answer

(a) P(X=6)≈0.128, P(X≥6)≈0.254 (b) Mode=4 (c) E=4.2, Var=2.73, SD=1.65 (d) P(Y=5)≈0.097, E(Y)≈2.86

Q21

Continuous probability: f(x) = kx(4−x) for 0≤x≤4, f(x)=0 otherwise

(a) Show k=3/32

(b) Find E(X) and Var(X)

(c) Find P(1 < X < 3)

(d) Find the median m such that ∫₀ᵐ f(x)dx=0.5.

Answer

(a) k=3/32 shown (1 < X < 3) (b) E(X)=2, Var=0.8 (c) 11/16=0.6875 (d) m=2

Q22

Confidence intervals and hypothesis: A sample of 40 light bulbs has mean lifetime x̄=1200hrs, s=150hrs

(a) Find a 99% confidence interval for μ

(b) The manufacturer claims μ=1250hrs. Test this at 5% (two-tailed).

(c) What sample size n is needed for margin of error ≤20hrs at 95% confidence?

(d) A second sample of 35 bulbs gives x̄=1180, s=160. Test if the two samples have equal means at 10% level.

Answer

(a) (1139, 1261) (b) Reject H₀ (c) n≥217 (d) Fail to reject; means not significantly different

Q23

Combined probability: Events A, B, C: P(A)=0.5, P(B|A)=0.6, P(B|A')=0.3, P(C|A∩B)=0.8, P(C|A∩B')=0.4, P(C|A'∩B)=0.5, P(C|A'∩B')=0.1.

(a) Find P(B) using total probability

(b) Find P(A|B) using Bayes

(c) Find P(C)

(d) Find P(A|C)

(e) [HL] If events are repeated independently 10 times, find P(C occurs at least 6 times).

Answer

(a) 0.45 (b) 2/3 (c) 0.43 (d) 0.744 (e) ≈0.197

Q24

Non-linear regression: data suggests model of form y=ax^b. After log transformation: ln(y)=ln(a)+b·ln(x). Data: (x,y): (1,3),(2,8.5),(3,15.6),(4,24),(5,33.7),(6,44),(7,55.8),(8,68).

(a) Transform data to (ln x, ln y)

(b) Find the linear regression of ln y on ln x via GDC

(c) Determine a and b

(d) Use model to predict y when x=10

(e) Calculate r for the transformed data and assess fit.

Answer

(a) Transform done (b) ln y≈0.97+1.50·ln x (c) a≈2.64, b≈1.50 (d) y≈83.5 (e) r≈0.9998

UNIT 4: STATISTICS & PROBABILITY