ZIMSEC A Level Statistics Paper 4 November 2008

Question 101

[1 marks]statistics - mean and standard deviation

A stem-and-leaf diagram of pocket money (in dollars) received by a group of girls is given below, with key $3|30 = \$3.30$ :

Stem 0: 50 50 50 75
Stem 1: 00 00 00 50 75
Stem 2: 00 00 00 50 50
Stem 3: 00 25 30 75
Stem 4: 50
Stem 5: 50

What is the mean pocket money received, to the nearest cent?

A$1.95
B$2.04
C$2.14
D$2.25

Question 102

[1 marks]statistics - mean and standard deviation

Using the same pocket money data (20 girls, values in cents: 50,50,50,75,100,100,100,150,175,200,200,200,250,250,300,325,330,375,450,550, mean \$2.14), what is the standard deviation, to the nearest cent?

A$1.37
B$1.41
C$1.52
D$2.54

Question 201

[1 marks]probability - conditional probability

A school's lower sixth intake is 55% from its own O-level pupils and 45% from other schools. Of those who did O-level elsewhere, 90% pass A-level; of those who did O-level at the school, 70% pass. A recent A-level graduate is selected at random. What is the probability that this pupil passed A-level?

A0.385
B0.405
C0.79
D0.81

Question 202

[1 marks]probability - conditional probability

Using the same school (55% own O-level pupils, 45% from elsewhere; pass rates 70% and 90% respectively, overall pass probability 0.79), given that a graduate passed A-level, what is the probability that they did O-level outside the school?

A0.45
B0.487
C0.513
D0.90

Question 301

[1 marks]probability - discrete random variables

A die has three faces numbered 0, two faces numbered 3, and one face numbered 6. It is thrown twice and

Y

is the product of the two scores shown. What is

P(Y=18)

A2/9
B1/9
C1/18
D1/2

Question 302

[1 marks]probability - discrete random variables

For the same die (faces 0,0,0,3,3,6), a boy pays \$5 to throw the die twice and win an amount equal to the product

Y

of the two scores, where

P(Y=0)=3/4

P(Y=9)=1/9

P(Y=18)=1/9

and

P(Y=36)=1/36

. What is the boy's expected profit or loss per game, and is the game fair to him?

Ano expected gain or loss; the game is fair
Ban expected gain of \$1; the game favours the boy
Can expected loss of \$1; the game favours the house, so it is not fair to the boy
Dan expected loss of \$2; the game favours the house, so it is not fair to the boy

Question 401

[1 marks]probability distributions - binomial and normal approximation

A library's books are 60% fiction. In a random collection of 6 books, what is the probability of getting 5 or more fiction books, correct to 3 decimal places?

A0.047
B0.187
C0.233
D0.767

Question 402

[1 marks]probability distributions - binomial and normal approximation

In the same library (60% fiction, 40% non-fiction), using a normal approximation, what is the probability that a random collection of 200 books contains exactly 80 non-fiction books?

A0
B0.045
C0.058
D0.115

Question 501

[1 marks]normal distribution - linear combinations

Machine A produces steel rods with length

N(15, 0.5)

and machine B produces rods with length

N(16, 0.1)

(mean, variance in cm). If two rods are selected at random from machine A's production, what are the mean and variance of their combined (total) length?

Amean 15, variance 1.0
Bmean 30, variance 0.5
Cmean 30, variance 2.0
Dmean 30, variance 1.0

Question 502

[1 marks]normal distribution - linear combinations

Two rods are selected at random from machine A's production (combined length mean 30, variance 1.0) and two from machine B's (combined length mean 32, variance 0.2). What is the probability that the total length of the two rods from machine B exceeds the total length of the two rods from machine A?

A0.034
B0.952
C0.966
D0.977

Question 601

[1 marks]probability distributions - Poisson

In a population, 1 in 1000 people are affected by a severe medical condition each year. For a company with 500 employees, the number of claims

X_A

received by the insurer in a year is modelled by a Poisson distribution. What is the mean of this distribution?

A0.05
B0.5
C5
D500

Question 602

[1 marks]probability distributions - Poisson

Company A has 500 employees, and its number of medical insurance claims per year is modelled by

X_A\sim\text{Po}(0.5)

. What is

P(X_A\ge2)

A0.090
B0.394
C0.910
D0.960

Question 603

[1 marks]probability distributions - Poisson

Company A (500 employees) has claims

X_A\sim\text{Po}(0.5)

and company B (750 employees) has claims

X_B\sim\text{Po}(0.75)

, independently. What is the probability that the total number of claims from both companies combined is exactly 2?

A0.076
B0.133
C0.209
D0.224

Question 701

[1 marks]sampling distributions - normal

Large samples of size

n

are taken from

N(100,225)

. Given that 95% of the sample means are less than 105, estimate

n

Question 702

[1 marks]sampling distributions - normal

X\sim N(4,9)

and

Y\sim N(5,16)

are independent. A sample of 20 observations is taken from

X

and a sample of 25 from

Y

. What is

P(\bar{Y}>\bar{X})

A0.169
B0.579
C0.831
D0.841

Question 801

[1 marks]hypothesis testing

A population has mean 9.27 and standard deviation 1.40. A sample of 36 has mean 8.39. Testing at the 1% level whether the population mean has decreased, which of the following gives the correct test statistic and conclusion?

A $z=-3.77$ ; reject $H_0$ , evidence at 1% that the mean has decreased
B $z=-0.88$ ; reject $H_0$ , evidence at 1% that the mean has decreased
C $z=-3.77$ ; do not reject $H_0$ , no evidence at 1% that the mean has decreased
D $z=-22.63$ ; reject $H_0$ , evidence at 1% that the mean has decreased

Question 802

[1 marks]hypothesis testing

An animal breeder claims a species has mean length 44 cm. A sample of 21 animals gives

\bar{x}=42

cm and

s=6

cm. Testing at the 5% level whether this refutes the breeder's claim, which of the following gives the correct test statistic and conclusion?

A $t=-7.00$ ; reject $H_0$ , evidence at 5% that the claim is false
B $t=-0.33$ ; do not reject $H_0$ , no evidence at 5% to refute the claim
C $t=-1.53$ ; do not reject $H_0$ , no evidence at 5% to refute the claim
D $t=-1.53$ ; reject $H_0$ , evidence at 5% that the claim is false

Question 901

[1 marks]regression and correlation

For the data of

P

(police cars: 2,3,4,4,5,6,6) and

N

(accidents: 45,40,36,42,30,25,24) recorded over 7 weeks, through which point must the regression line of

P

N

pass?

A(30, 242)
B(34.6, 4.3)
C(4.3, 34.6)
D(36, 4)

Question 902

[1 marks]regression and correlation

Using the data of

P

(police cars) and

N

(accidents) over 7 weeks (

\sum N=242

\sum P=30

\sum NP=966

\sum N^2=8786

), the regression line of

P

N

P=10.15-0.170N

. What value does this line predict for

P

when

N=35

A4.2
B5.9
C10.1
D16.1

Question 903

[1 marks]regression and correlation

For the data of

P

(police cars) and

N

(accidents) over 7 weeks, with

S_{NN}\approx419.7

S_{PP}\approx13.4

and

S_{NP}\approx-71.1

, what is the product-moment correlation coefficient, and what does it indicate?

A $r\approx-0.95$ ; a weak correlation, N and P are largely unrelated
B $r\approx-0.95$ ; a strong negative correlation, more patrol cars associated with fewer accidents
C $r\approx+0.95$ ; a strong positive correlation, more patrols associated with more accidents
D $r\approx0.90$ ; a very strong positive correlation

Question 1001

[1 marks]chi-squared goodness of fit test

In a seed viability test, 600 seeds were planted in 100 rows of 6, with the number germinating per row given below.

Seeds germinating	0	1	2	3	4	5	6
Observed rows	1	4	7	29	33	18	8

What is the mean number of seeds germinating per row?

A3.65
B3.75
C3.79
D4.75

Question 1002

[1 marks]chi-squared goodness of fit test

For the same seed data (mean 3.75 seeds germinating per row out of 6), a binomial model

B(6,p)

with the same mean is fitted, giving

p=0.625

. What is the expected number of rows (out of 100) with exactly 3 seeds germinating?

A1.3
B19.3
C25.8
D27.6

Question 1003

[1 marks]chi-squared goodness of fit test

For the seed germination data, after combining classes with expected frequency below 5 (giving observed frequencies 12, 29, 33, 18, 8 against expected frequencies 14.6, 25.8, 32.2, 21.5, 6.0 for a fitted

B(6,0.625)

model), which of the following gives the correct

\chi^2

statistic, degrees of freedom, and conclusion at the 5% level?

A $\chi^2\approx34.3$ , df $=3$ , critical value 7.815; reject $H_0$ , the data do not fit a binomial model
B $\chi^2\approx2.16$ , df $=4$ , critical value 9.49; do not reject $H_0$
C $\chi^2\approx2.16$ , df $=3$ , critical value 7.815; reject $H_0$ , the data do not fit a binomial model
D $\chi^2\approx2.16$ , df $=3$ , critical value 7.815; do not reject $H_0$ , the data are consistent with a binomial model

Statistics Paper 4 November 2008

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The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 401

Question 402

Question 501

Question 502

Question 601

Question 602

Question 603

Question 701

Question 702

Question 801

Question 802

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

More sittings of this paper

The answers, and why they are the answers

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