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ZIMSEC A Level · 6046/1 · N2022

Statistics Paper 1 November 2022

Questions
29
Total marks
152
Syllabus code
6046/1

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Questions
29
Pass mark
18
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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]normal distribution
The volume of dish washing liquid in a satchet is normally distributed with mean 400 ml and standard deviation 45 ml. The volume of fabric softener in a bottle is normally distributed with mean 650 ml and standard deviation 50 ml, independently of the satchets. Find the probability that the total volume of 4 randomly chosen satchets is less than the total volume of 2 randomly chosen bottles.
  1. A0.0044
  2. B0.0228
  3. C0.1587
  4. D0.5000

Question 201

[1 marks]normal distribution
A machine packs washing powder into packets with mean weight 2 kg. The weights are normally distributed, and 10% of packets weigh less than 1.95 kg. Find the standard deviation of the packet weights, correct to 3 decimal places (in kg).

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Question 202

[1 marks]normal distribution
A machine packs washing powder into packets that are normally distributed with mean 2 kg and standard deviation 0.039 kg. Find the proportion of packets that weigh more than 2.10 kg.
  1. A0.26%
  2. B0.52%
  3. C1.05%
  4. D10.00%

Question 301

[1 marks]continuous random variables
A continuous random variable X has probability density function f(x)=38x2f(x) = \dfrac{3}{8}x^2 for 0≤x≤20 \leq x \leq 2. Find the value of the cumulative distribution function F(1)F(1).

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Question 302

[1 marks]continuous random variables
A continuous random variable X has probability density function f(x)=38x2f(x) = \dfrac{3}{8}x^2 for 0≤x≤20 \leq x \leq 2, with cumulative distribution function F(x)=x3/8F(x) = x^3/8. Find the median mm of X.
  1. A1.260
  2. B1.414
  3. C1.587
  4. D2.000

Question 401

[1 marks]normal distribution
A random variable Y is normally distributed with mean μ\mu and standard deviation σ\sigma. Given that P(Y>34)=0.0228P(Y > 34) = 0.0228 and P(Y<25)=0.0062P(Y < 25) = 0.0062, find the value of μ\mu.

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Question 402

[1 marks]normal distribution
A random variable Y is normally distributed with mean μ\mu and standard deviation σ\sigma. Given that P(Y>34)=0.0228P(Y > 34) = 0.0228 and P(Y<25)=0.0062P(Y < 25) = 0.0062, find the value of σ\sigma.

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Question 501

[1 marks]confidence intervals
Ten measurements of the burning time x (in hours) of a candle gave ∑(x−5)=12.5\sum(x-5)=12.5 and ∑(x−5)2=23.8\sum(x-5)^2=23.8. Find the sample mean burning time, xˉ\bar{x}.

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Question 502

[1 marks]confidence intervals
Ten measurements of the burning time x (in hours) of a candle gave ∑(x−5)=12.5\sum(x-5)=12.5 and ∑(x−5)2=23.8\sum(x-5)^2=23.8. Using the t-distribution, find a 95% confidence interval for the population mean burning time.
  1. A(5.57, 6.93)
  2. B(4.09, 8.41)
  3. C(5.15, 7.35)
  4. D(5.66, 6.84)

Question 601

[1 marks]normal distribution
The time Mrs Moyo spends in a clinic is normally distributed with mean 25 minutes and standard deviation 4 minutes. Find the probability that she is in the clinic for more than 28 minutes.

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Question 602

[1 marks]normal distribution
Mrs Moyo's travel time to a clinic is normally distributed with mean 15 minutes and standard deviation 2 minutes, independently of the time she spends in the clinic, which is normally distributed with mean 25 minutes and standard deviation 4 minutes. Find the probability that on a particular day she spends more time travelling than in the clinic.
  1. A0.0127
  2. B0.0250
  3. C0.0478
  4. D0.2266

Question 701

[1 marks]probability, Poisson distribution
Alan and Alex take turns firing at a target, with Alan firing first. Each has probability 0.6 of hitting the target on any shot, independently. The first to hit wins the game. Find the probability that Alan hits the target on his 2nd shot (i.e. Alan misses shot 1, Alex misses his turn, then Alan hits).

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Question 702

[1 marks]probability, Poisson distribution
The number of spelling errors per page in a document follows a Poisson distribution with mean 0.3. Find the probability that a randomly chosen page contains no spelling errors.
  1. A0.2592
  2. B0.3000
  3. C0.7408
  4. D0.9048

Question 703

[1 marks]probability, Poisson distribution
The number of spelling errors per page in a document follows a Poisson distribution with mean 0.3. Find the probability that a randomly chosen page contains two or more spelling errors.
  1. A0.0333
  2. B0.0370
  3. C0.2592
  4. D0.9963

Question 704

[1 marks]probability, Poisson distribution
The number of spelling errors per page in a document follows a Poisson distribution with mean 0.3, independently from page to page. Find the probability that, out of three randomly chosen pages, the third page is the first one to contain a spelling error.
  1. A0.0174
  2. B0.1423
  3. C0.1920
  4. D0.4269

Question 705

[1 marks]probability, Poisson distribution
Alan and Alex take turns firing at a target, with Alan firing first. Each has probability 0.6 of hitting the target on any shot, independently, and the first to hit wins. Find the probability that Alan wins the game.

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Question 801

[1 marks]cumulative frequency, statistics
A sample of 80 steel rods was measured to the nearest mm, and the shortest length class recorded was 92-93 mm. What is the least possible actual length, in mm, of a steel rod in this class?

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Question 802

[1 marks]cumulative frequency, statistics
A sample of 80 steel rods was measured to the nearest mm, and the longest length class recorded was 102-103 mm. What is the greatest possible actual length, in mm, of a steel rod in this class?

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Question 803

[1 marks]cumulative frequency, statistics
A sample of 80 steel rods (measured to the nearest mm) gave this frequency distribution: 92-93 mm: 2 rods, 94-95 mm: 7 rods, 96-97 mm: 18 rods, 98-99 mm: 37 rods, 100-101 mm: 12 rods, 102-103 mm: 4 rods. Using the class boundaries (91.5, 93.5, 95.5, 97.5, 99.5, 101.5, 103.5 mm) and linear interpolation on the cumulative frequencies, estimate the median length in mm.

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Question 901

[1 marks]Poisson distribution, chi-squared goodness of fit
The table shows the number of telephone calls received per 10-minute period over 8 hours: 0 calls in 8 periods, 1 call in 19 periods, 2 calls in 26 periods, 3 calls in 13 periods, 4 calls in 7 periods, 5 calls in 5 periods, 6 calls in 1 period, 7 calls in 1 period, 8 calls in 0 periods. Find the mean number of calls per 10-minute period.

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Question 902

[1 marks]Poisson distribution, chi-squared goodness of fit
The table shows the number of telephone calls received per 10-minute period over 8 hours: 0 calls in 8 periods, 1 call in 19 periods, 2 calls in 26 periods, 3 calls in 13 periods, 4 calls in 7 periods, 5 calls in 5 periods, 6 calls in 1 period, 7 calls in 1 period, 8 calls in 0 periods. Find the total number of telephone calls received over the 8 hours.

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Question 1001

[1 marks]time series
In statistics, what name is given to a set of data values recorded at successive points in time (for example, daily, weekly, or yearly)?

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Question 1002

[1 marks]time series
Daily egg sales (in crates) at a supermarket were: Mon=2, Tue=5, Wed=4, Thu=4, Fri=9, Sat=7, Sun=6, Mon=8. Calculate the 3-point moving average centred on Wednesday (i.e. the average of the Tuesday, Wednesday and Thursday values).

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Question 1101

[1 marks]probability, conditional probability
In a group of students, 63% are boys and the probability that a boy studies Mathematics is 19\dfrac{1}{9}. Find the probability that a randomly selected student from this group is a boy who studies Mathematics.

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Question 1102

[1 marks]probability, conditional probability
In a group of students, 63% are boys and 37% are girls. The probability that a boy studies Mathematics is 19\dfrac{1}{9} and the probability that a girl studies Mathematics is 27\dfrac{2}{7}. Find the probability that a randomly selected student from this group is NOT studying Mathematics.

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Question 1103

[1 marks]probability, conditional probability
In a group of students, 63% are boys and 37% are girls. The probability that a girl studies Mathematics is 27\dfrac{2}{7} and the probability that a boy studies Mathematics is 19\dfrac{1}{9}. Given that a randomly selected Mathematics student is being considered, find the probability that this student is female.
  1. A0.2857
  2. B0.3700
  3. C0.6015
  4. D0.7000

Question 1201

[1 marks]regression and correlation
For 10 students, Sociology (x) and Economics (y) percentage marks gave n=10n=10, ∑x=557\sum x=557, ∑y=661\sum y=661, ∑x2=34377\sum x^2=34377, ∑y2=47323\sum y^2=47323, ∑xy=40280\sum xy=40280. Calculate the product moment correlation coefficient between x and y.

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Question 1202

[1 marks]regression and correlation
For 10 students, Sociology (x) and Economics (y) percentage marks gave n=10n=10, ∑x=557\sum x=557, ∑y=661\sum y=661, ∑x2=34377\sum x^2=34377, ∑y2=47323\sum y^2=47323, ∑xy=40280\sum xy=40280. Find the gradient of the regression line of y on x.
  1. A0.95
  2. B0.97
  3. C1.03
  4. D1.17

Question 1203

[1 marks]regression and correlation
The regression line of Economics mark (y) on Sociology mark (x) for a group of students is y=8.59+1.03xy = 8.59 + 1.03x. Use this line to estimate the Economics mark for a student who scored 57% in Sociology.

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