ZIMSEC A Level Statistics Paper 4 June 2009

Question 101

[1 marks]probability

An experiment has only two possible outcomes. The first outcome occurs with probability

p

and the second outcome occurs with probability

p^2

. Find the value of

p

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 102

[1 marks]probability

A survey in a city shows that the probability a person is in favour of capital punishment is 0.55 and the probability a person is against it is 0.45. If two people are selected at random, find the probability that at least one of them favours capital punishment.

A0.2025
B0.3025
C0.5500
D0.7975

Question 201

[1 marks]continuous random variables

The continuous random variable

X

has probability density function

f(x)=\frac{1}{28}(6x-4)

for

2\le x\le4

(and 0 otherwise). Find

E(X)

, giving your answer as a fraction or correct to 3 decimal places.

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Question 301

[1 marks]cumulative frequency distribution

The table below shows the masses of 200 students measured to the nearest kg.

Mass (kg)	Number of students
46-50	20
51-55	60
56-60	56
61-65	35
66-70	19
71-75	10

Using upper class boundaries, what is the cumulative number of students with mass 60.5 kg or less?

A80
B136
C171
D190

Question 302

[1 marks]cumulative frequency distribution

For the same 200 students (masses grouped to the nearest kg as 46-50: 20, 51-55: 60, 56-60: 56, 61-65: 35, 66-70: 19, 71-75: 10), use the cumulative frequency curve to estimate the interquartile range of the masses.

A10 kg
B9.5 kg
C53 kg
D62.5 kg

Question 303

[1 marks]cumulative frequency distribution

For the same 200 students (masses grouped to the nearest kg as 46-50: 20, 51-55: 60, 56-60: 56, 61-65: 35, 66-70: 19, 71-75: 10), use the cumulative frequency curve to estimate the percentage of students with a mass of 54 kg or less.

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Question 401

[1 marks]normal distribution

The heights of flowers in a bed are normally distributed with mean 21.1 cm and standard deviation 4.0 cm. Find the probability that a randomly chosen flower has a height greater than 25 cm.

A0.165
B0.309
C0.404
D0.835

Question 402

[1 marks]normal distribution

The heights of flowers in a bed are normally distributed with mean 21.1 cm and standard deviation 4.0 cm. Find the probability that a randomly chosen flower has a height less than 25 cm, correct to 3 decimal places.

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Question 501

[1 marks]discrete probability distribution and expectation

A trader buys 250 cabbages at $5 000 each and sells them at $10 000 each. Any cabbages remaining at the end of the day are sold to a local farmer at $1 000 each. If exactly 100 cabbages are demanded that day, find her profit (or loss) for the day.

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Question 502

[1 marks]discrete probability distribution and expectation

A trader buys 250 cabbages at $5 000 each and sells them at $10 000 each, with any leftover cabbages sold to a local farmer at $1 000 each. If demand that day is 300 cabbages (more than the 250 she bought), find her profit for the day.

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Question 503

[1 marks]discrete probability distribution and expectation

A trader buys 250 cabbages at $5 000 each and sells them at $10 000 each, with any leftover cabbages sold to a local farmer at $1 000 each. Daily demand follows: 0 cabbages with probability 0.1, 100 with probability 0.5, 200 with probability 0.3, and 300 with probability 0.1. Find her expected profit for the day.

A-$215 000
B$125 000
C$215 000
D$237 500

Question 601

[1 marks]poisson distribution

Vehicles pass a highway service station near Marondera towards Mutare at an average rate of 10 per minute, and in the opposite direction at an average rate of 12 per minute, each following a Poisson distribution. Find the probability that a total of 4 vehicles (both directions combined) pass in a given period of 15 seconds.

A0.0102
B0.134
C0.156
D0.168

Question 602

[1 marks]poisson distribution

Vehicles pass a highway service station near Marondera towards Mutare at an average rate of 10 per minute, and in the opposite direction at an average rate of 12 per minute, each following a Poisson distribution. Find the probability that no vehicles (either direction) pass in a given period of 30 seconds.

A $4.09\times10^{-3}$
B $2.74\times10^{-10}$
C $4.54\times10^{-5}$
D $1.67\times10^{-5}$

Question 701

[1 marks]chi-square goodness of fit test

The table below shows the number of absentees in a class of 50 college students on each day of a given week.

Days of the week	Monday	Tuesday	Wednesday	Thursday	Friday
Number of absentees	14	7	4	9	16

Testing the null hypothesis that absentees are equally distributed over the five days (expected frequency 10 per day), calculate the chi-square test statistic.

A2.0
B9.8
C10.34
D98.0

Question 702

[1 marks]chi-square goodness of fit test

Testing whether absentees are equally distributed over the five days of the week (Monday 14, Tuesday 7, Wednesday 4, Thursday 9, Friday 16 absentees out of 50 students, expected frequency 10 per day), the calculated chi-square statistic is 9.8 with 4 degrees of freedom, and the critical value at the 5% significance level is

\chi^2_{4,0.05}=9.488

. What is the correct conclusion?

AReject $H_0$ , but only because the degrees of freedom should be taken as 5, not 4.
BReject $H_0$ ; there is significant evidence that absentees are not equally distributed over the days.
CThe test is inconclusive because 9.8 is close to the critical value of 9.488.
DFail to reject $H_0$ ; there is insufficient evidence that absentees are not equally distributed over the days.

Question 801

[1 marks]central limit theorem and sampling distributions

State the name of the theorem which says that, for a sufficiently large sample size, the sampling distribution of the sample mean is approximately normal with mean

\mu

and standard deviation

\sigma/\sqrt n

, regardless of the shape of the population's own distribution.

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Question 802

[1 marks]central limit theorem and sampling distributions

Electricity bills for households in a city have a probability distribution with mean $65 and standard deviation $25. For a random sample of 75 households, find the standard error of the sample mean, correct to 2 decimal places.

A$0.33
B$2.89
C$8.66
D$25.00

Question 803

[1 marks]central limit theorem and sampling distributions

Electricity bills for households in a city have a probability distribution with mean $65 and standard deviation $25. For a random sample of 75 households, find the probability that the sample mean exceeds the population mean by at least $5 (i.e. the sample mean is at least $70), correct to 3 decimal places or 4 significant figures.

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Question 901

[1 marks]correlation and regression

Six people's annual incomes and the sums assured of their life policies (both in thousands of dollars) are: (52,250), (58,300), (31,100), (43,150), (65,500), (24,75). What best describes the relationship between annual income and sum assured shown by these data?

AA strong positive relationship: sum assured tends to increase as annual income increases.
BA weak relationship, with sum assured largely independent of annual income.
CA strong negative relationship: sum assured tends to decrease as annual income increases.
DNo clear relationship between annual income and sum assured.

Question 902

[1 marks]correlation and regression

Six people's annual incomes

x

and the sums assured

y

of their life policies (both in thousands of dollars) are: (52,250), (58,300), (31,100), (43,150), (65,500), (24,75). Find the gradient of the regression line of sum assured on annual income, correct to 2 decimal places.

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Question 903

[1 marks]correlation and regression

Six people's annual incomes

x

and the sums assured

y

of their life policies (both in thousands of dollars) are: (52,250), (58,300), (31,100), (43,150), (65,500), (24,75). Using the regression line of sum assured on annual income fitted to these data, estimate the sum assured for a person with an annual income of $45 000, to the nearest $1 000.

Answer this when you sit the paper.

Question 1001

[1 marks]poisson distribution

A randomly chosen counsellor deals with cases of a broken family at an average rate of 1 per year, each case independent of others. Taking a month as a twelfth of a year, find the combined mean number of cases (i.e.

\lambda

) that three such counsellors together attend to in a period of one month.

Answer this when you sit the paper.

Question 1002

[1 marks]poisson distribution

A randomly chosen counsellor deals with cases of a broken family at an average rate of 1 per year, each case independent of others. Taking a month as a twelfth of a year, find the probability that three such counsellors together attend to no cases of a broken family in a period of one month.

A0.050
B0.368
C0.779
D0.920

Question 1003

[1 marks]poisson distribution

A randomly chosen counsellor deals with cases of a broken family at an average rate of 1 per year, each case independent of others and of the other counsellors. Find the probability that three such counsellors together attend to at least 3 cases of a broken family in a period of one year, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1101

[1 marks]confidence intervals and hypothesis testing

The standard deviation of the mass of a certain component is 0.3 g. Using a 99% confidence level (z = 2.576), find the number of components that must be sampled so that the sample mean is within 0.06 g of the true population mean.

Answer this when you sit the paper.

Question 1102

[1 marks]confidence intervals and hypothesis testing

The standard deviation of the mass of a certain component is 0.3 g, known from past data. When constructing a confidence interval for the population mean mass using the formula

\bar x\pm z\sigma/\sqrt n

, what must be assumed about the population of component masses?

AThe sample must be selected with replacement from an infinite population.
BThe sample size must exceed 30 for the interval to be valid.
CThe population of component masses is normally distributed.
DThe population standard deviation is unknown and must be estimated from the sample.

Question 1103

[1 marks]confidence intervals and hypothesis testing

The Tax Department randomly sampled 10 tax returns with interest deductions

2984,

2910,

3050,

3333,

3101,

3002,

3415,

1897,

2416,

3872 (sample mean $2998, sample standard deviation approximately $539.80). Testing

H_0:\mu=3011

against

H_1:\mu\ne3011

at the 5% significance level, the calculated test statistic is

t\approx-0.076

and the two-tailed critical value with 9 degrees of freedom is

t_{9,0.025}=2.262

. What is the correct conclusion?

AFail to reject $H_0$ ; there is insufficient evidence that the 1999 average interest deduction differs from the 1993 average of $3011.
BFail to reject $H_0$ , but only because the degrees of freedom should be taken as 10, not 9.
CThe test cannot be completed because the sample mean $2998 is numerically less than the hypothesised mean $3011.
DReject $H_0$ ; there is significant evidence that the 1999 average interest deduction differs from the 1993 average of $3011.

Statistics Paper 4 June 2009

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The questions

Question 101

Question 102

Question 201

Question 301

Question 302

Question 303

Question 401

Question 402

Question 501

Question 502

Question 503

Question 601

Question 602

Question 701

Question 702

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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