ZIMSEC A Level Statistics Paper 4 June 2007

Question 101

[1 marks]conditional probability / Bayes theorem

A manufacturing plant uses three machines, A, B and C, in its production process. The total daily output contributions of machines A, B and C are 40%, 45% and 15% respectively. It is known that 4% of the tins produced by A are defective, 3% of those produced by B are defective, and 1% of those produced by C are defective. What is the probability that one tin chosen at random from the day's production is defective?

A $0.027$
B $0.031$
C $0.032$
D $0.034$

Question 102

[1 marks]conditional probability / Bayes theorem

A manufacturing plant uses three machines, A, B and C, contributing 40%, 45% and 15% of daily output respectively, with defect rates of 4%, 3% and 1% for A, B and C respectively. Given that a randomly chosen tin is defective, what is the probability, to 3 decimal places, that it came from machine B?

Answer this when you sit the paper.

Question 201

[1 marks]normal distribution / linear combinations

A factory produces two types of nut and bolt with normally distributed masses. Type A bolts have mean mass 20.5 g and Type A nuts have mean mass 5 g; Type B bolts have mean mass 20 g and Type B nuts have mean mass 4.7 g. Each bolt is fitted with two nuts. What is the mean, in grams, of (total mass of a Type A bolt-and-nuts unit) minus (total mass of a Type B bolt-and-nuts unit)?

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Question 202

[1 marks]normal distribution / linear combinations

A factory produces two types of nut and bolt with independent, normally distributed masses, each with standard deviation 0.2 g. Type A bolts have mean 20.5 g and Type A nuts have mean 5 g; Type B bolts have mean 20 g and Type B nuts have mean 4.7 g. Each bolt is fitted with two nuts. What is the probability that the total mass of a Type A bolt-and-nuts unit is greater than the total mass of a Type B bolt-and-nuts unit?

A $0.988$
B $0.994$
C $0.997$
D $0.999$

Question 301

[1 marks]discrete probability distribution

Given that

P(X=x) = k\left(\dfrac{1}{3}\right)^x

for

x=1,2,3,4

, where

k

is a constant chosen so the probabilities sum to 1, what is the value of

k

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Question 302

[1 marks]discrete probability distribution

Given that

P(X=x) = \dfrac{81}{40}\left(\dfrac{1}{3}\right)^x

for

x=1,2,3,4

, what is

P(X\leq3)

A $0.025$
B $0.481$
C $0.900$
D $0.975$

Question 303

[1 marks]discrete probability distribution

Given that

P(X=x) = \dfrac{81}{40}\left(\dfrac{1}{3}\right)^x

for

x=1,2,3,4

, what is

E(X)

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Question 401

[1 marks]descriptive statistics / stem and leaf

The marks obtained by 23 students in a Physics test, from a stem-and-leaf diagram (key: 6 | 3 means 63), are: Stem 2, leaf 2 5; Stem 3, leaf 4 7 7; Stem 4, leaf 3 8 8; Stem 5, leaf 1 6 7 7 7 8; Stem 6, leaf 3 4; Stem 7, leaf 6 8 9 9; Stem 8, leaf 3 5; Stem 9, leaf 1. What is the median mark?

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Question 402

[1 marks]descriptive statistics / stem and leaf

Q_1

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Question 403

[1 marks]descriptive statistics / stem and leaf

A $14$
B $17.5$
C $21$
D $35$

Question 501

[1 marks]normal distribution

Nails produced by a machine have normally distributed lengths. Given that 2% of the nails are longer than 10.4 cm and 95% of the nails have lengths between 10.3 cm and 10.4 cm, what is the standard deviation of the distribution, to 4 decimal places?

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Question 502

[1 marks]normal distribution

Nails produced by a machine have normally distributed lengths. Given that 2% of the nails are longer than 10.4 cm and 95% of the nails have lengths between 10.3 cm and 10.4 cm, what is the mean length of the distribution, to 4 decimal places?

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Question 601

[1 marks]Poisson distribution

Weekly supply to car dealer B follows a Poisson distribution with mean

\frac{2}{3}

cars. What is the probability that, in a given week, fewer than three cars are supplied to B?

A $0.856$
B $0.920$
C $0.970$
D $0.995$

Question 602

[1 marks]Poisson distribution

Weekly supply to car dealer A follows a Poisson distribution with mean

\frac{4}{3}

cars, and weekly supply to car dealer B independently follows a Poisson distribution with mean

\frac{2}{3}

cars. What is the probability that, in a given week, fewer than three cars in total are supplied to A and B combined?

A $0.406$
B $0.677$
C $0.857$
D $0.920$

Question 603

[1 marks]Poisson distribution

For a random variable

X

representing the number of cars supplied to a dealer each week,

X\sim\text{Po}(\lambda)

. What value of

\lambda

satisfies

P(X=0)=P(X=2)

Answer this when you sit the paper.

Question 701

[1 marks]continuous probability distribution / exponential

The mileage

X

(in thousands of km) obtained from a brand of car tyre has probability density function

f(x)=ke^{-x/8}

for

x>0

, and

f(x)=0

otherwise, where

k

is a constant. What is the value of

k

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Question 702

[1 marks]continuous probability distribution / exponential

The mileage

X

(in thousands of km) obtained from a brand of car tyre has probability density function

f(x)=\frac18 e^{-x/8}

for

x>0

, and

f(x)=0

otherwise. What is the mean mileage of a tyre, in km?

Answer this when you sit the paper.

Question 703

[1 marks]continuous probability distribution / exponential

The mileage

X

(in thousands of km) obtained from a brand of car tyre has probability density function

f(x)=\frac18 e^{-x/8}

for

x>0

, and

f(x)=0

otherwise. What is the probability, to 4 decimal places, that a tyre lasts at most 100 000 km?

Answer this when you sit the paper.

Question 801

[1 marks]discrete distributions / central limit theorem

Five cards are numbered 2, 2, 4, 4, 6. Two cards are drawn with replacement, and

X

is the sum of the two numbers drawn. What is

P(X=8)

A $0.16$
B $0.24$
C $0.32$
D $0.40$

Question 802

[1 marks]discrete distributions / central limit theorem

Five cards are numbered 2, 2, 4, 4, 6. Two cards are drawn with replacement, and

X

is the sum of the two numbers drawn. What is

E(X)

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Question 803

[1 marks]discrete distributions / central limit theorem

Five cards are numbered 2, 2, 4, 4, 6. Two cards are drawn with replacement, and

X

is the sum of the two numbers drawn, with

E(X)=7.2

and

\text{Var}(X)=4.48

. For a random sample of 30 such draws, what is the probability that the sample mean exceeds 7.5?

A $0.219$
B $0.444$
C $0.500$
D $0.781$

Question 901

[1 marks]regression and correlation

A social scientist recorded ARD rate (

X\%

) and divorce rate (

Y\%

) over 6 periods, with

n=6

\sum X=684

\sum Y=64

\sum X^2=92312

\sum Y^2=926

\sum XY=9106

. What is the product moment correlation coefficient, to 3 decimal places?

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Question 902

[1 marks]regression and correlation

A social scientist recorded ARD rate (

X\%

) and divorce rate (

Y\%

) over 6 periods, with

n=6

\sum X=684

\sum Y=64

\sum X^2=92312

\sum Y^2=926

\sum XY=9106

. What is the gradient of the regression line of

y

upon

x

, to 3 decimal places?

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Question 903

[1 marks]regression and correlation

Given the regression line of divorce rate

y

(%) upon ARD rate

x

(%) is

y=-3.731+0.1263x

, what is the estimated divorce rate, to 1 decimal place, for an ARD rate of 300%?

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Question 1001

[1 marks]hypothesis testing / central limit theorem

A driver records the times (minutes) taken to complete a 50 km lap over 30 laps: Time 23 (frequency 3), 24 (7), 25 (8), 26 (6), 27 (3), 28 (2), 29 (1). What is the sample mean lap time, to 1 decimal place?

Answer this when you sit the paper.

Question 1002

[1 marks]hypothesis testing / central limit theorem

A sample of 30 lap times has mean 25.3 minutes and standard deviation 1.218 minutes. Testing

H_0:\mu=25

against

H_1:\mu>25

, what is the value of the test statistic?

A $-1.35$
B $0.25$
C $1.35$
D $1.65$

Question 1003

[1 marks]hypothesis testing / central limit theorem

A test of

H_0:\mu=25

against

H_1:\mu>25

, based on a sample of 30 lap times with mean 25.3 minutes, gives a test statistic

z=1.35

, compared against the 5% one-tailed critical value

z=1.645

. What should be concluded?

ADo not reject $H_0$ ; there is insufficient evidence at the 5% level that the mean lap time exceeds 25 minutes.
BReject $H_0$ ; there is sufficient evidence at the 5% level that the mean lap time is below 25 minutes.
CDo not reject $H_0$ ; the test is inconclusive because the sample of 30 laps is too small.
DReject $H_0$ ; there is sufficient evidence at the 5% level that the mean lap time exceeds 25 minutes.

Question 1101

[1 marks]chi-squared goodness of fit test

A sample of 400 people tasted 4 brands of herbal tea, with 95, 104, 109 and 92 people respectively favouring brands P, Q, R and S. Testing the hypothesis that preference is uniformly distributed across the 4 brands (so each has an expected frequency of 100), what is the value of the chi-squared test statistic?

A $1.86$
B $3.72$
C $7.44$
D $18.6$

Question 1102

[1 marks]chi-squared goodness of fit test

A random sample of 100 plants from a breeding experiment gives 20 red, 35 pink and 45 white flowers. The hypothesised proportions are

\frac14

red,

\frac14

pink and

\frac12

white, giving expected frequencies of 25, 25 and 50 respectively. What is the value of the chi-squared test statistic?

A $1.5$
B $2.75$
C $4.5$
D $5.5$

Question 1103

[1 marks]chi-squared goodness of fit test

A random sample of 100 plants from a breeding experiment gives 20 red, 35 pink and 45 white flowers, with hypothesised proportions

\frac14

red,

\frac14

pink and

\frac12

white. This gives a chi-squared test statistic of 5.5 with 2 degrees of freedom. What is the least significance level

\alpha

(as a percentage, to 1 decimal place) at which the null hypothesis can be rejected?

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Statistics Paper 4 June 2007

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The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 501

Question 502

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 801

Question 802

Question 803

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1101

Question 1102

Question 1103

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The answers, and why they are the answers

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