ZIMSEC A Level Statistics Paper 1 November 2020

Question 101

[1 marks]conditional probability

A student travels to school by bus, car, or on foot with probabilities

\frac{1}{6}

\frac{1}{3}

, and

\frac{1}{2}

respectively. The probability of being late is

\frac{1}{5}

if by bus,

\frac{1}{4}

if by car, and

\frac{1}{20}

if on foot. What is the probability that the student is early (not late) for school?

A $\frac{103}{120}$
B $\frac{1}{2}$
C $\frac{19}{40}$
D $\frac{17}{120}$

Question 102

[1 marks]conditional probability

A student travels to school by bus, car, or on foot with probabilities

\frac{1}{6}

\frac{1}{3}

, and

\frac{1}{2}

respectively. The probability of being late is

\frac{1}{5}

if by bus,

\frac{1}{4}

if by car, and

\frac{1}{20}

if on foot. Given that the student arrives early on a particular day, what is the probability, as a fraction in lowest terms, that the student travelled on foot?

Answer this when you sit the paper.

Question 201

[1 marks]descriptive statistics / box plots

The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the median time, in minutes?

Answer this when you sit the paper.

Question 202

[1 marks]descriptive statistics / box plots

The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the interquartile range (Q3 minus Q1) of these times, in minutes?

A3.5
B4.0
C4.5
D5.5

Question 203

[1 marks]descriptive statistics / box plots

The times to the nearest minute taken by 12 students to complete a task are 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44, giving Q1 = 42, median = 44.5, and Q3 = 46.5 minutes (IQR = 4.5). The upper fence for outliers is Q3 + 1.5×IQR = 53.25 minutes. What can be concluded about the value 55 and the overall shape of the distribution?

A55 is below the upper fence, so it is not an outlier, and the distribution is symmetric
B55 exceeds the upper fence, so it is an outlier, and the distribution is positively skewed
C55 exceeds the upper fence, so it is an outlier, and the distribution is negatively skewed
D55 is below the upper fence, so it is not an outlier, and the distribution is positively skewed

Question 301

[1 marks]combinations and probability

A team of four people is chosen at random from 5 women and 6 men (11 people in total), with no restrictions on composition. In how many ways can the team be chosen?

A165
B210
C330
D462

Question 302

[1 marks]combinations and probability

A team of four is chosen at random from 5 women and 6 men, with the restriction that there must be more men than women (so either 3 men and 1 woman, or 4 men and 0 women). In how many ways can such a team be chosen?

A15
B100
C115
D135

Question 303

[1 marks]combinations and probability

A team of four is chosen at random from 5 women and 6 men (11 people). What is the probability that the team contains exactly one man (and three women)?

A2/11
B6/11
C3/11
D1/11

Question 401

[1 marks]discrete random variables

A discrete random variable X has probability distribution

P(X=1)=0.1

P(X=2)=0.3

P(X=a)=0.4

P(X=b)=0.2

, where

a

and

b

are distinct values different from 1 and 2. Given

E(X)=3.5

and

\text{Var}(X)=2.65

, what is the value of

a

Answer this when you sit the paper.

Question 402

[1 marks]discrete random variables

A discrete random variable X has probability distribution

P(X=1)=0.1

P(X=2)=0.3

P(X=a)=0.4

P(X=b)=0.2

, where

a

and

b

are distinct values different from 1 and 2. Given

E(X)=3.5

and

\text{Var}(X)=2.65

, what is the value of

b

Answer this when you sit the paper.

Question 501

[1 marks]regression

For the data

x

: 25, 30, 35, 40, 45, 50 and

y

: 78, 70, 65, 58, 48, 42, what is the gradient of the regression line of

y

x

, correct to 2 decimal places?

Answer this when you sit the paper.

Question 502

[1 marks]regression

For the data

x

: 25, 30, 35, 40, 45, 50 and

y

: 78, 70, 65, 58, 48, 42, which of the following is the regression line of

y

x

A $\hat{y} = 114.4 + 1.45x$
B $\hat{y} = 60.2 - 1.45x$
C $\hat{y} = 1.45 - 114.4x$
D $\hat{y} = 114.4 - 1.45x$

Question 503

[1 marks]regression

A regression line of

y

x

is fitted to the data

x

: 25, 30, 35, 40, 45, 50 and

y

: 78, 70, 65, 58, 48, 42, giving

\hat{y} \approx 114.4 - 1.45x

. Can a value of

x

be estimated for

y = 54

, and why?

ANo, because $y=54$ lies outside the range of the observed $y$ -values, so this would be extrapolation
BYes, because $y=54$ lies within the range of the observed $y$ -values (42 to 78), so this is interpolation
CYes, because a regression line predicts $x$ from $y$ just as reliably outside the observed data
DNo, because a regression equation cannot be solved algebraically for $x$ given $y$

Question 601

[1 marks]summary statistics

The masses of 18 pupils are summarised by

\sum x = 745

and

\sum x^2 = 33951

(in kg). What is the mean mass, to 1 decimal place?

Answer this when you sit the paper.

Question 602

[1 marks]summary statistics

The masses of 18 pupils are summarised by

\sum x = 745

and

\sum x^2 = 33951

(kg). What is the variance of the masses, to 1 decimal place?

A13.2
B41.4
C173.1
D1886.2

Question 603

[1 marks]summary statistics

The masses of 18 pupils are summarised by

\sum x = 745

kg. One pupil leaves the group, and the mean mass of the remaining 17 pupils is exactly 41 kg. What is the mass, in kg, of the pupil who left?

Answer this when you sit the paper.

Question 604

[1 marks]summary statistics

The masses of 18 pupils are summarised by

\sum x = 745

kg and

\sum x^2 = 33951

kg². One pupil of mass 48 kg leaves, leaving 17 pupils with mean mass 41 kg. What is the standard deviation of the remaining 17 pupils' masses, to 2 decimal places?

A13.44 kg
B14.02 kg
C180.59 kg
D12.98 kg

Question 701

[1 marks]binomial distribution

Given

X \sim \text{Bin}(5, \frac{1}{6})

, what is

P(X=2)

, correct to 3 decimal places?

A0.003
B0.032
C0.161
D0.402

Question 702

[1 marks]binomial distribution

Given

X \sim \text{Bin}(5, \frac{1}{6})

, what is

P(X \geq 4)

, correct to 4 decimal places?

A0.0001
B0.0033
C0.0161
D0.4019

Question 703

[1 marks]binomial distribution

Given

X \sim \text{Bin}(5, \frac{1}{6})

, what is

P(X<3)

, correct to 4 decimal places?

Answer this when you sit the paper.

Question 801

[1 marks]geometric distribution

X \sim \text{Geo}(0.6)

(X = trial number of the first success, with success probability 0.6), what is

P(X=4)

Answer this when you sit the paper.

Question 802

[1 marks]geometric distribution

X \sim \text{Geo}(0.6)

, what are

E(X)

and

\text{Var}(X)

A $E(X)=0.6$ , $\text{Var}(X)=0.24$
B $E(X)=0.667$ , $\text{Var}(X)=1.111$
C $E(X)=1.667$ , $\text{Var}(X)=0.667$
D $E(X)=1.667$ , $\text{Var}(X)=1.111$

Question 803

[1 marks]geometric distribution

X \sim \text{Geo}(0.6)

, what is

P(X \geq 3)

Answer this when you sit the paper.

Question 901

[1 marks]continuous random variables / pdf

A random variable X has pdf

f(x) = c(2x-x^2)

for

0 < x < \frac{3}{2}

, and

f(x)=0

otherwise, where

c>0

. What is the value of

c

, as a fraction in lowest terms?

Answer this when you sit the paper.

Question 902

[1 marks]continuous random variables / pdf

A random variable X has pdf

f(x) = \frac{8}{9}(2x-x^2)

for

0 < x < \frac{3}{2}

, and

f(x)=0

otherwise. What is

E(X)

, as a fraction in lowest terms?

Answer this when you sit the paper.

Question 1001

[1 marks]normal distribution

X is normally distributed with mean

\mu=3

and variance

\sigma^2=9

. What is

P(2<X<5)

, correct to 3 decimal places?

A0.369
B0.378
C0.622
D0.748

Question 1002

[1 marks]normal distribution

X is normally distributed with mean

\mu=3

and variance

\sigma^2=9

. What is

P(X>0)

, correct to 4 decimal places?

Answer this when you sit the paper.

Question 1003

[1 marks]normal distribution

X is normally distributed with mean

\mu=3

and variance

\sigma^2=9

. What is

P(|X-3|>6)

, correct to 4 decimal places?

A0.0228
B0.0456
C0.1587
D0.9544

Question 1101

[1 marks]time series / trend equations

A trend equation is given as

\hat{y} = 284 + 14.4x

, where the origin is 1 July 1974, the

x

unit is 1 year, and the

y

unit is annual sales ×

1{,}000

. Using

x=11

for the year 1985, what is the projected annual sales, in dollars, for 1985?

Answer this when you sit the paper.

Question 1102

[1 marks]time series / trend equations

A trend equation is given as

\hat{y} = 284 + 14.4x

, where

x

is in years and

y

is in units of annual sales ×

1{,}000

. What is the annual dollar increase in sales indicated by this trend equation?

Answer this when you sit the paper.

Question 1201

[1 marks]continuous random variables / binomial

The lifetime

X

(in hours) of an electronic device has pdf

f(x) = \frac{10}{x^2}

for

x>10

, and

f(x)=0

otherwise. What is

P(X<20)

Answer this when you sit the paper.

Question 1202

[1 marks]continuous random variables / binomial

The lifetime

X

(hours) of an electronic device has pdf

f(x) = \frac{10}{x^2}

for

x>10

, and

f(x)=0

otherwise. What is the cumulative distribution function

F(x)

for

x>10

A $F(x) = 1 - \frac{10}{x^2}$
B $F(x) = \frac{x}{10} - 1$
C $F(x) = 1 - \frac{10}{x}$
D $F(x) = \frac{10}{x} - 1$

Question 1203

[1 marks]continuous random variables / binomial

The lifetime

X

(hours) of a type of electronic device has pdf

f(x) = \frac{10}{x^2}

for

x>10

(zero otherwise), so

P(X \leq 20) = 0.5

. For 6 independently chosen devices of this type, what is the probability that at least 3 of them function for at most 20 hours, as a fraction in lowest terms?

Answer this when you sit the paper.

Question 1301

[1 marks]normal distribution / linear combinations

A \sim N(75, 36)

. What is

P(A>65)

, correct to 4 decimal places?

Answer this when you sit the paper.

Question 1302

[1 marks]normal distribution / linear combinations

A \sim N(75, 36)

and

B \sim N(65, 25)

are independent. Three independent values of

A

are chosen and three independent values of

B

are chosen. What is the probability that the mean of the three

A

-values exceeds the mean of the three

B

-values, correct to 3 decimal places?

A0.500
B0.953
C0.987
D0.999

Question 1303

[1 marks]normal distribution / linear combinations

A \sim N(75, 36)

and

B \sim N(65, 25)

are independent. What is

P(A > 1.15B)

, correct to 3 decimal places?

Answer this when you sit the paper.

Question 1401

[1 marks]normal approximation to binomial

In a sample of 500 randomly selected injuries, where 40% of all injuries occur at home, use a normal approximation with continuity correction to find

P(X=190)

, where

X

is the number occurring at home, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1402

[1 marks]normal approximation to binomial

In a sample of 500 randomly selected injuries, where 40% of all injuries occur at home, use a normal approximation to find

P(180 \leq X \leq 210)

, where

X

is the number occurring at home, correct to 3 decimal places.

A0.024
B0.397
C0.801
D0.963

Question 1403

[1 marks]normal approximation to binomial

In a sample of 500 randomly selected injuries, where 40% of all injuries occur at home, use a normal approximation with continuity correction to find

P(X>180)

, where

X

is the number occurring at home, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1501

[1 marks]chi-squared goodness of fit / Poisson distribution

In a chi-squared goodness-of-fit test of whether the number of calls per 5-minute interval (600 intervals observed) follows a Poisson distribution with mean 2.5, what is the expected frequency for exactly 0 calls, to 2 decimal places, using

E(r) = 600 \times e^{-2.5} \times \frac{2.5^r}{r!}

Answer this when you sit the paper.

Question 1502

[1 marks]chi-squared goodness of fit / Poisson distribution

A chi-squared goodness-of-fit test checks whether the number of calls per 5-minute interval (600 intervals observed) follows a Poisson distribution with a specified mean of 2.5 (not estimated from the data). After combining classes so no expected frequency is too small, there are 7 classes (0, 1, 2, 3, 4, 5, and 6-or-more). How many degrees of freedom does the test statistic have?

Question 1503

[1 marks]chi-squared goodness of fit / Poisson distribution

A chi-squared goodness-of-fit test compares the observed frequencies 34, 131, 160, 136, 72, 37, and 30 (for 6-or-more, after combining classes) from 600 five-minute intervals against a Poisson(2.5) model. What is the calculated chi-squared test statistic, correct to 2 decimal places?

Answer this when you sit the paper.

Question 1504

[1 marks]chi-squared goodness of fit / Poisson distribution

Testing at the 1% significance level whether call frequencies at a switchboard (600 five-minute intervals) follow a Poisson distribution with mean 2.5, the calculated chi-squared statistic is 7.90 with 6 degrees of freedom, and the critical value

\chi^2_6

at the 1% level is 16.81. What is the conclusion?

ADo not reject $H_0$ , because the calculated statistic exceeds the critical value
BReject $H_0$ , because a sample of 600 is too large for a chi-squared test
CDo not reject $H_0$ : there is insufficient evidence that the data does not follow a Poisson(2.5) distribution
DReject $H_0$ : there is sufficient evidence that the data does not follow a Poisson(2.5) distribution

Statistics Paper 1 November 2020

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The questions

Question 101

Question 102

Question 201

Question 202

Question 203

Question 301

Question 302

Question 303

Question 401

Question 402

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 801

Question 802

Question 803

Question 901

Question 902

Question 1001

Question 1002

Question 1003

Question 1101

Question 1102

Question 1201

Question 1202

Question 1203

Question 1301

Question 1302

Question 1303

Question 1401

Question 1402

Question 1403

Question 1501

Question 1502

Question 1503

Question 1504

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