ZIMSEC A Level Statistics Paper 1 November 2018

Question 101

[1 marks]descriptive statistics

A group of twenty people played a game. The frequency distribution of their scores is: score 1 with frequency 2, score 2 with frequency 5, score 4 with frequency 7, and score

x

with frequency 6. If the mean score is 5, what is the value of

x

Answer this when you sit the paper.

Question 102

[1 marks]descriptive statistics

A group of twenty people played a game with scores 1, 2, 4 and 10, having frequencies 2, 5, 7 and 6 respectively (mean score 5). What is the variance of this distribution?

Answer this when you sit the paper.

Question 201

[1 marks]conditional probability

A head of school can contact parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively, using only one method. The probability that parents receive the message is 0.6 if e-mail is used, 0.8 if letter is used and 1 if cellphone is used. What is the probability that the parents receive the message?

Answer this when you sit the paper.

Question 202

[1 marks]conditional probability

A head of school contacts parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively. The probability parents receive the message is 0.6 for e-mail, 0.8 for letter and 1 for cellphone, so the overall probability of receiving the message is 0.82. Given that the parents received the message, what is the probability that it was sent by e-mail?

A0.240
B0.293
C0.400
D0.600

Question 301

[1 marks]time series

Which of the following is not one of the four traditional components of a time series?

ACorrelation
BSeasonal variation
CCyclical variation
DTrend

Question 302

[1 marks]time series

Compared with a 3-point moving average, a 4-point moving average of the same data produces a plot that is...

Asmoother, and its plotted points align exactly with the original time periods
Bless smooth, with more random fluctuation retained
Cidentical, since both remove all seasonal variation
Dsmoother, and its plotted points fall between the original time periods

Question 401

[1 marks]permutations and combinations

Julius Caesar is one of the novels in a collection of 20 novels. A learner is going to choose 5 of these novels to take on holiday. In how many ways can the learner choose the 5 novels?

A4845
B11628
C15504
D1860480

Question 402

[1 marks]permutations and combinations

From a collection of 20 novels, including Julius Caesar, a learner chooses 5 novels to take on holiday. How many of the possible choices of 5 novels include Julius Caesar?

A969
B3876
C4845
D11628

Question 403

[1 marks]permutations and combinations

In how many ways can 3 boys and 4 girls stand in a line if the 3 boys must stand next to each other, as one block?

Answer this when you sit the paper.

Question 501

[1 marks]continuous random variables

A continuous random variable X has probability density function

f(x)=\frac{3}{32}(4-x^2)

for

-2\le x\le2

and

f(x)=0

otherwise. What is

E(X)

Answer this when you sit the paper.

Question 502

[1 marks]continuous random variables

A continuous random variable X has probability density function

f(x)=\frac{3}{32}(4-x^2)

for

-2\le x\le2

and

f(x)=0

otherwise, with

E(X)=0

. What is

\text{Var}(X)

Answer this when you sit the paper.

Question 601

[1 marks]Poisson distribution

Under which conditions can the Poisson distribution be used as an approximation to the Binomial distribution?

Awhen $n$ is large and $p$ is small
Bwhen $n$ is large and $p$ is close to 1
Cwhen $n$ and $p$ are both large
Dwhen $n$ is small and $p$ is close to 0.5

Question 602

[1 marks]Poisson distribution

Potato seeds are packed in packets of 200 seeds, with on average 2% of the seeds in a packet rotten. A packet is substandard if it contains 5 or more rotten seeds. Using a Poisson approximation with mean

\lambda=200\times0.02=4

, what is the probability that a packet is substandard, correct to 3 decimal places?

Answer this when you sit the paper.

Question 701

[1 marks]sampling distributions

Given that

E(X)=1.2

and

\text{Var}(X)=0.6

, and

Y=10X+50

, what is

E(Y)

Answer this when you sit the paper.

Question 702

[1 marks]sampling distributions

Given that

E(X)=1.2

and

\text{Var}(X)=0.6

, and

Y=10X+50

, what is

\text{Var}(Y)

Answer this when you sit the paper.

Question 703

[1 marks]sampling distributions

The random variable Y has mean 62 and variance 60. A random sample of 160 independent observations of Y is taken. What is the probability that the sample mean lies between 61 and 62.5?

A0.207
B0.742
C0.793
D0.949

Question 801

[1 marks]normal distribution

In a normal distribution with mean

\mu

and standard deviation

\sigma

P(X>5.6)=0.5

. What is the value of

\mu

Answer this when you sit the paper.

Question 802

[1 marks]normal distribution

In a normal distribution with mean

\mu=5.6

and standard deviation

\sigma

P(X>4.8)=0.6554

. What is the value of

\sigma

Answer this when you sit the paper.

Question 803

[1 marks]normal distribution

X is normally distributed with

P(X>4.8)=0.6554

. If 4 independent observations of X are taken, what is the probability that at least 2 of them are greater than 4.8?

A0.014
B0.107
C0.121
D0.879

Question 901

[1 marks]hypothesis testing

A random sample of 80 pockets of manure gives

\sum x=79.53

and

\sum x^2=100.4621

(in kg). Testing

H_0:\mu=1.10

against

H_1:\mu<1.10

, what is the value of the test statistic

t

, to 2 decimal places?

Answer this when you sit the paper.

Question 902

[1 marks]hypothesis testing

A test of

H_0:\mu=1.10

kg against

H_1:\mu<1.10

kg for the mass of manure pockets gives a test statistic of

t=-1.82

with 79 degrees of freedom, compared against a one-tailed 5% critical value of

-1.664

. What is the correct conclusion?

AReject $H_0$ : there is evidence at the 5% level that the mean mass is less than 1.10 kg
BDo not reject $H_0$ : there is no evidence at the 5% level that the mean mass is less than 1.10 kg
CReject $H_0$ : there is evidence at the 5% level that the mean mass is greater than 1.10 kg
DDo not reject $H_0$ : the sample provides no information about the population mean

Question 1001

[1 marks]descriptive statistics

The heights (cm) of 15 children are: 115, 120, 158, 132, 125, 104, 142, 160, 145, 104, 162, 117, 107, 124, 134. What is the median height?

Answer this when you sit the paper.

Question 1002

[1 marks]descriptive statistics

The heights (cm) of 15 children are: 115, 120, 158, 132, 125, 104, 142, 160, 145, 104, 162, 117, 107, 124, 134. What is the interquartile range of these heights?

Answer this when you sit the paper.

Question 1003

[1 marks]descriptive statistics

The heights (cm) of 15 children are: 115, 120, 158, 132, 125, 104, 142, 160, 145, 104, 162, 117, 107, 124, 134, with lower quartile 115, median 125 and upper quartile 145, so there is more spread above the median (20) than below it (10). What does this suggest about the distribution?

AIt is negatively skewed, with more spread among the lower heights
BIt is positively skewed, with more spread among the higher heights
CIt is symmetric, since the median lies exactly between the quartiles
DIt has no meaningful skew, since sample size is too small to tell

Question 1101

[1 marks]probability

Events A and B are such that

P(A\cup B)=0.9

P(A\cap B)=0.2

and

P(A|B)=0.8

. What is

P(B)

Answer this when you sit the paper.

Question 1102

[1 marks]probability

Events A and B are such that

P(A\cup B)=0.9

P(A\cap B)=0.2

and

P(B)=0.25

. What is

P(A')

, the probability of the complement of A?

Answer this when you sit the paper.

Question 1103

[1 marks]probability

An unbiased die is thrown repeatedly until a six appears. What is the expected number of throws needed?

Answer this when you sit the paper.

Question 1201

[1 marks]estimation and confidence intervals

In statistics, what is the term for a subset selected from a population, used to make inferences about that population?

Aa census
Ba parameter
Ca sample
Da population

Question 1202

[1 marks]estimation and confidence intervals

The amount of pocket money ($X) received by 120 learners was summarised by

\sum(X-50)=-221

. What is the unbiased estimate of the population mean amount of pocket money?

Answer this when you sit the paper.

Question 1203

[1 marks]estimation and confidence intervals

The amount of pocket money ($X) received by 120 learners was summarised by

\sum(X-50)=-221

and

\sum(X-50)^2=4708

. What is the unbiased estimate of the population standard deviation, to 2 decimal places?

Answer this when you sit the paper.

Question 1204

[1 marks]estimation and confidence intervals

The amount of pocket money ($X) received by 120 learners has unbiased mean estimate 48.16 and unbiased standard deviation estimate 6.01. What is the 95% confidence interval for the population mean, to 2 decimal places?

A(46.75, 49.57)
B(47.26, 49.06)
C(48.16, 49.24)
D(47.08, 49.23)

Question 1301

[1 marks]chi-squared goodness of fit

The number of calls per hour to a hotline over 7 one-hour periods (9am-4pm) totalled 931 calls: 132, 151, 143, 129, 117, 134, 125. Under the null hypothesis that calls follow a uniform distribution across the 7 periods, what is the expected number of calls in each period?

Answer this when you sit the paper.

Question 1302

[1 marks]chi-squared goodness of fit

The observed numbers of calls in 7 one-hour periods are 132, 151, 143, 129, 117, 134, 125 (expected 133 per period under a uniform distribution). What is the value of the

\chi^2

test statistic, to 2 decimal places?

Answer this when you sit the paper.

Question 1303

[1 marks]chi-squared goodness of fit

\chi^2

goodness-of-fit test for whether hourly call numbers follow a uniform distribution over 7 periods gives a test statistic of 5.73 with 6 degrees of freedom, against a 10% critical value of 10.645. What is the correct conclusion?

AReject $H_0$ : the calls definitely follow a uniform distribution
BDo not reject $H_0$ : the test is inconclusive and should be repeated with more data
CDo not reject $H_0$ : there is insufficient evidence at the 10% level that the calls are not uniformly distributed
DReject $H_0$ : there is evidence at the 10% level that the calls are not uniformly distributed

Question 1401

[1 marks]correlation and regression

Six sales representatives made the following numbers of calls to potential customers: 7, 6, 8, 6, 1, 2, with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. What is the product moment correlation coefficient between number of calls and sales turnover, to 2 decimal places?

Answer this when you sit the paper.

Question 1402

[1 marks]correlation and regression

Six sales representatives made calls (7, 6, 8, 6, 1, 2) with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. Using the least squares regression equation of turnover on number of calls, what is the gradient (slope) of the regression line, to 3 decimal places?

Answer this when you sit the paper.

Question 1403

[1 marks]correlation and regression

Using the regression equation of sales turnover ($1000) on number of calls,

y=7.292+0.675x

, fitted to six sales representatives' data, what is the predicted turnover ($1000) for a representative who made 4 calls, to 1 decimal place?

Answer this when you sit the paper.

Question 1404

[1 marks]correlation and regression

A regression equation of sales turnover on number of calls was fitted using data for six representatives who made between 1 and 8 calls each. Why is it not appropriate to use this equation to predict turnover for a representative who made 10 calls?

Athe correlation coefficient is too low for the equation to be used for any prediction
B10 calls falls outside the range of the data used to fit the equation, so the linear relationship may not extend that far
Cthe regression equation only applies to whole numbers of calls below 8
Da value of 10 calls would necessarily produce a negative predicted turnover value

Statistics Paper 1 November 2018

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The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 401

Question 402

Question 403

Question 501

Question 502

Question 601

Question 602

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 1103

Question 1201

Question 1202

Question 1203

Question 1204

Question 1301

Question 1302

Question 1303

Question 1401

Question 1402

Question 1403

Question 1404

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