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ZIMSEC A Level · 6046/2 · N2019

Statistics Paper 2 November 2019

Questions
36
Total marks
152
Time allowed
180 min
Syllabus code
6046/2

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Questions
36
Pass mark
22
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Answer every question in the printed order, get marked at the end, then see the answers.

The questions

Question 101

[1 marks]continuous random variables / probability density functions
A continuous random variable XX has probability density function f(x)=x12f(x) = \dfrac{x}{12} for 0≤x<30 \le x < 3, f(x)=k(x−8)f(x) = k(x-8) for 3≤x≤83 \le x \le 8, and f(x)=0f(x) = 0 otherwise, where kk is a constant. Since the total area under ff must equal 1, what is the value of kk?
  1. A-0.10
  2. B-0.05
  3. C-0.04
  4. D0.05

Question 102

[1 marks]continuous random variables / probability density functions
A continuous random variable XX has probability density function f(x)=x12f(x) = \dfrac{x}{12} for 0≤x<30 \le x < 3 and f(x)=−120(x−8)f(x) = -\dfrac{1}{20}(x-8) for 3≤x≤83 \le x \le 8 (zero otherwise). What is E(X)E(X), correct to 2 decimal places?
  1. A3.53
  2. B3.60
  3. C3.67
  4. D3.75

Question 103

[1 marks]continuous random variables / probability density functions
A continuous random variable XX has probability density function f(x)=x12f(x) = \dfrac{x}{12} for 0≤x<30 \le x < 3 and f(x)=−120(x−8)f(x) = -\dfrac{1}{20}(x-8) for 3≤x≤83 \le x \le 8 (zero otherwise). What is the median of XX, correct to 2 decimal places?

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Question 201

[1 marks]descriptive statistics / grouped data
A method of collecting statistical data involves obtaining information from every single member of a population, rather than from a sample of it. What is this method of data collection called?

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Question 202

[1 marks]descriptive statistics / grouped data
In a survey of 100 athletes at a marathon, the amount of water taken, in litres, was recorded as: 0-0.5 litres by 8 athletes, 0.5-1.0 litres by 20 athletes, 1.0-1.5 litres by 29 athletes, 1.5-2.0 litres by 22 athletes, and 2.0-2.5 litres by 21 athletes. Using the midpoint of each class, what is the estimated mean amount of water taken, in litres?
  1. A1.30
  2. B1.35
  3. C1.39
  4. D1.45

Question 203

[1 marks]descriptive statistics / grouped data
In a survey of 100 athletes at a marathon, the amount of water taken, in litres, was recorded as: 0-0.5 litres by 8 athletes, 0.5-1.0 litres by 20 athletes, 1.0-1.5 litres by 29 athletes, 1.5-2.0 litres by 22 athletes, and 2.0-2.5 litres by 21 athletes. Using the midpoint of each class and a mean of 1.39 litres, what is the estimated standard deviation of the amount of water taken, correct to 3 decimal places?

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Question 301

[1 marks]discrete random variables / expectation and variance
A discrete random variable XX has distribution P(X=2)=0.1P(X=2)=0.1, P(X=3)=0.4P(X=3)=0.4, P(X=4)=0.1P(X=4)=0.1, P(X=5)=0.3P(X=5)=0.3, P(X=6)=0.1P(X=6)=0.1. What is E(X)E(X)?
  1. A3.5
  2. B3.9
  3. C4.0
  4. D4.1

Question 302

[1 marks]discrete random variables / expectation and variance
A discrete random variable XX has distribution P(X=2)=0.1P(X=2)=0.1, P(X=3)=0.4P(X=3)=0.4, P(X=4)=0.1P(X=4)=0.1, P(X=5)=0.3P(X=5)=0.3, P(X=6)=0.1P(X=6)=0.1. What is Var(X)\text{Var}(X)?
  1. A1.41
  2. B1.49
  3. C1.59
  4. D1.69

Question 303

[1 marks]discrete random variables / expectation and variance
A discrete random variable XX has distribution P(X=2)=0.1P(X=2)=0.1, P(X=3)=0.4P(X=3)=0.4, P(X=4)=0.1P(X=4)=0.1, P(X=5)=0.3P(X=5)=0.3, P(X=6)=0.1P(X=6)=0.1, so that E(X)=3.9E(X)=3.9 and Var(X)=1.49\text{Var}(X)=1.49. If Y=3X−2Y = 3X - 2, what is Var(Y)\text{Var}(Y)?

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Question 401

[1 marks]Poisson distribution
The number of people who use a lift in a minute follows a Poisson distribution with mean 2. What is the probability that exactly 3 people use the lift in a minute, correct to 3 decimal places?
  1. A0.090
  2. B0.135
  3. C0.180
  4. D0.271

Question 402

[1 marks]Poisson distribution
The number of people who use a lift follows a Poisson distribution with mean 2 per minute. In a 2-minute period the mean becomes 4. What is the probability that fewer than 4 people use the lift in this 2-minute period, correct to 3 decimal places?
  1. A0.238
  2. B0.434
  3. C0.566
  4. D0.629

Question 403

[1 marks]Poisson distribution
The number of people who use a lift follows a Poisson distribution with mean 2 per minute. In a 3-minute period the mean becomes 6. What is the probability that more than 2 people use the lift in this 3-minute period, correct to 3 decimal places?
  1. A0.062
  2. B0.849
  3. C0.938
  4. D0.983

Question 501

[1 marks]permutations and conditional probability
How many 3-digit code numbers can be formed using the digits 1, 2, 3, 4 and 5, if the order of the digits matters and repetition of digits is permitted?
  1. A15
  2. B60
  3. C120
  4. D125

Question 502

[1 marks]permutations and conditional probability
How many 3-digit code numbers can be formed using the digits 1, 2, 3, 4 and 5, if the order of the digits matters and no digit may be repeated?
  1. A10
  2. B20
  3. C60
  4. D125

Question 503

[1 marks]permutations and conditional probability
At a school, the probability that a learner passes Advanced Level is 0.8. The probability that a learner who passes Advanced Level proceeds to Tertiary Education is 0.9. The probability that a learner who fails Advanced Level does not proceed to Tertiary Education is 0.4 (so the probability that a learner who fails Advanced Level does proceed to Tertiary Education is 0.6). Given that a learner proceeds to Tertiary Education, what is the probability that the learner failed Advanced Level, correct to 3 decimal places?

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Question 601

[1 marks]regression and correlation
For a set of paired data on temperature θ\theta (°C) and time tt (minutes), the regression line of θ\theta on tt is θ=25.94+0.899t\theta = 25.94 + 0.899t. Using this equation, what is the estimated value of θ\theta when t=65t = 65 minutes, correct to 1 decimal place?
  1. A81.4
  2. B84.4
  3. C87.4
  4. D90.4

Question 602

[1 marks]regression and correlation
For a set of paired data on temperature θ\theta (°C) and time tt (minutes), the regression line of θ\theta on tt is θ=25.94+0.899t\theta = 25.94 + 0.899t. Using this equation, what is the value of θ\theta when t=95t = 95 minutes, correct to 1 decimal place?
  1. A105.3
  2. B108.3
  3. C111.3
  4. D114.3

Question 603

[1 marks]regression and correlation
For 8 paired observations of tt and θ\theta, the product moment correlation coefficient is calculated to be r≈0.975r \approx 0.975. What does this value indicate about the linear relationship between tt and θ\theta?
  1. AA weak positive linear correlation.
  2. BNo linear correlation.
  3. CA very strong positive linear correlation.
  4. DA very strong negative linear correlation.

Question 701

[1 marks]chi-squared goodness of fit / Poisson distribution
A farmer counted the number of weeds in 100 randomly chosen square-metre plots, with results: 0 weeds (18 plots), 1 weed (25 plots), 2 weeds (25 plots), 3 weeds (16 plots), 4 weeds (7 plots), 5 weeds (9 plots). What is the mean number of weeds per square metre?

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Question 702

[1 marks]chi-squared goodness of fit / Poisson distribution
A chi-squared goodness-of-fit test is used to test whether data follow a Poisson distribution. After combining categories so that no expected frequency is below 5, there are 5 categories of data remaining, and the Poisson mean used in the test was estimated from the sample itself. How many degrees of freedom does the test statistic have?
  1. A2
  2. B3
  3. C4
  4. D5

Question 703

[1 marks]chi-squared goodness of fit / Poisson distribution
In a chi-squared goodness-of-fit test for whether the number of weeds per square metre follows a Poisson distribution, the calculated test statistic is χ2=2.095\chi^2 = 2.095 with 3 degrees of freedom, and the critical value at the 5% significance level is χ32=7.815\chi^2_{3} = 7.815. Since 2.095<7.8152.095 < 7.815, what should be concluded?
  1. ADo not reject H0; there is insufficient evidence that the number of weeds does not follow a Poisson distribution.
  2. BReject H0 because a sample of only 100 square-metre plots is too small to test this hypothesis.
  3. CReject H0; there is sufficient evidence that the number of weeds does not follow a Poisson distribution.
  4. DDo not reject H0; the number of weeds follows a Poisson distribution with complete certainty.

Question 801

[1 marks]normal distribution / linear combinations
The masses of hard cover books are normally distributed with mean 0.5 kg and standard deviation 0.15 kg. What is the probability that a randomly chosen hard cover book has a mass less than 0.65 kg, correct to 4 decimal places?
  1. A0.1587
  2. B0.6915
  3. C0.8413
  4. D0.9772

Question 802

[1 marks]normal distribution / linear combinations
The masses of exercise books are normally distributed with mean 0.2 kg and standard deviation 0.07 kg, independently of each other. What is the probability that the total mass of 4 randomly chosen exercise books is less than 0.9 kg, correct to 4 decimal places?
  1. A0.5753
  2. B0.6394
  3. C0.7625
  4. D0.9236

Question 803

[1 marks]normal distribution / linear combinations
The masses of hard cover books are normally distributed with mean 0.5 kg and standard deviation 0.15 kg. The masses of exercise books are normally distributed with mean 0.2 kg and standard deviation 0.07 kg, independently of hard cover books. What is the probability that the mass of a randomly chosen hard cover book is less than 3 times the mass of a randomly chosen exercise book, correct to 4 decimal places?
  1. A0.3492
  2. B0.5000
  3. C0.6508
  4. D0.7517

Question 901

[1 marks]estimation / hypothesis testing
A random sample of 75 bags of maize meal, each of mass xx kg, gave ∑(x−5)=738.5\sum(x-5)=738.5. What is the unbiased estimate of the mean mass, correct to 2 decimal places?

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Question 902

[1 marks]estimation / hypothesis testing
A random sample of 75 bags of maize meal, each of mass xx kg, gave ∑(x−5)=738.5\sum(x-5)=738.5 and ∑(x−5)2=18723\sum(x-5)^2=18723. What is the unbiased estimate of the variance of the mass, correct to 2 decimal places?

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Question 903

[1 marks]estimation / hypothesis testing
A sample of 100 packs of sugar had sample mean mass xˉ=1.98\bar x = 1.98 kg and sample variance S2=0.2846S^2 = 0.2846. What is the unbiased estimate of the population variance of the pack masses, correct to 4 decimal places?

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Question 904

[1 marks]estimation / hypothesis testing
A sample of 100 packs of sugar, nominally 2 kg each, had sample mean mass xˉ=1.98\bar x = 1.98 kg and unbiased population variance estimate 0.28750.2875 kg2^2. Testing H0:μ=2H_0: \mu = 2 against H1:μ<2H_1: \mu < 2 at the 10% significance level, the test statistic is z=1.98−20.2875/100≈−0.373z = \dfrac{1.98-2}{\sqrt{0.2875/100}} \approx -0.373, and the 10% one-tailed critical value is z=−1.282z = -1.282. What should be concluded?
  1. ADo not reject H0; the sample mean mass of 1.98 kg is exactly equal to the target mass of 2 kg.
  2. BDo not reject H0; there is insufficient evidence at the 10% significance level that the mean pack mass is below 2 kg.
  3. CReject H0 because the sample of 100 packs of sugar was not selected using proper random sampling.
  4. DReject H0; there is sufficient evidence at the 10% significance level that the mean pack mass is below 2 kg.

Question 1001

[1 marks]normal distribution / inverse normal
The heights hh metres of people in a community are normally distributed with mean μ\mu and standard deviation δ\delta. It is given that P(h<1.2)=0.15P(h<1.2)=0.15 and P(h>1.6)=0.10P(h>1.6)=0.10. What is the value of δ\delta, correct to 3 decimal places?
  1. A0.150
  2. B0.173
  3. C0.190
  4. D0.200

Question 1002

[1 marks]normal distribution / inverse normal
The heights hh metres of people in a community are normally distributed with mean μ\mu and standard deviation δ\delta. It is given that P(h<1.2)=0.15P(h<1.2)=0.15 and P(h>1.6)=0.10P(h>1.6)=0.10, from which δ≈0.173\delta \approx 0.173. What is the value of μ\mu, correct to 2 decimal places?

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Question 1003

[1 marks]normal distribution / inverse normal
The heights hh metres of people in a community are normally distributed with mean μ=1.38\mu = 1.38 m and standard deviation δ=0.173\delta = 0.173 m. What is P(∣h−1.38∣<0.1)P(|h-1.38|<0.1), correct to 3 decimal places?
  1. A0.219
  2. B0.438
  3. C0.562
  4. D0.876

Question 1101

[1 marks]binomial and geometric distributions / normal approximation
The probability that a form 3 learner passes a given test is 0.6. In a class of 15 form 3 learners, what is the probability that exactly 4 learners pass the test, correct to 4 decimal places?
  1. A0.0074
  2. B0.0136
  3. C0.0245
  4. D0.1268

Question 1102

[1 marks]binomial and geometric distributions / normal approximation
If X∼Geo(0.25)X \sim \text{Geo}(0.25), where XX is the number of trials up to and including the first success and the probability of success on each trial is 0.25, what is Var(X)\text{Var}(X)?
  1. A0.44
  2. B4
  3. C12
  4. D16

Question 1103

[1 marks]binomial and geometric distributions / normal approximation
If X∼Geo(0.25)X \sim \text{Geo}(0.25), where XX is the number of trials up to and including the first success and the probability of success on each trial is 0.25, what is P(X>3)P(X>3), correct to 4 decimal places?

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Question 1201

[1 marks]time series / moving averages
Using the seven consecutive daily sales values \$162, \$143, \$138, \$138, \$149, \$204 and \$90, what is the seven-day moving average, correct to the nearest whole number?
  1. A143
  2. B145
  3. C146
  4. D150

Question 1202

[1 marks]time series / moving averages
Using the seven consecutive daily sales values \$143, \$138, \$138, \$149, \$204, \$90 and \$155, what is the seven-day moving average, correct to the nearest whole number?
  1. A144
  2. B145
  3. C146
  4. D148

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