• Live Chat
  • Register
  • Login
Login with Facebook
ibworldacademy@gmail.com
+91-9818369374
Skype Me™!
Quick Contact

Statistics

1.       A sample of 70 batteries was tested to see how long they last. The results were:

Time (hours)

Number of batteries (frequency)

0 £ t < 10

2

10 £ t < 20

4

20 £ t < 30

8

30 £ t < 40

9

40 £ t < 50

12

50 £ t < 60

13

60 £ t < 70

8

70 £ t < 80

7

80 £ t < 90

6

90 £ t £100

1

Total

70

 

          Find

(a)     the sample standard deviation;

(b)     an unbiased estimate of the standard deviation of the population from which this sample is taken.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

2.       A machine fills bottles with orange juice.  A sample of six bottles is taken at random.  The bottles contain the following amounts (in ml) of orange juice: 753, 748, 749, 752, 750, 751.

          Find

(a)     the sample standard deviation;

(b)     an unbiased estimate of the population standard deviation from which this sample is taken.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

3.       A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown below.

Weight (g)

29.6

29.7

29.8

29.9

30.0

30.1

30.2

30.3

Frequency

2

3

4

5

7

5

3

1

          Find unbiased estimates of

(a)     the mean of the population from which this sample is taken;

(b)     the variance of the population from which this sample is taken.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

4.       The 80 applicants for a Sports Science course were required to run 800 metres and their times were recorded. The results were used to produce the following cumulative frequency graph.

          Estimate

(a)     the median;

(b)     the interquartile range.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

5.       Consider the six numbers, 2, 3, 6, 9, a and b. The mean of the numbers is 6 and the variance is 10. Find the value of a and of b, if a < b.

Working:

 

 

Answer:

..........................................................................

..........................................................................

(Total 6 marks)

 


 

6.       A teacher drives to school. She records the time taken on each of 20 randomly chosen days. She finds that

          = 626 and = 19780.8, where xi denotes the time, in minutes, taken on the ith day.

          Calculate an unbiased estimate of

(a)     the mean time taken to drive to school;

(b)     the variance of the time taken to drive to school.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

7.       A business man spends X hours on the telephone during the day. The probability density function of X is given by

f (x) =

(a)     (i)      Write down an integral whose value is E(X).

(ii)     Hence evaluate E(X).

(3)

(b)     (i)      Show that the median, m, of X satisfies the equation

          m4 – 16m2 + 24 = 0.

(ii)     Hence evaluate m.

(5)

(c)     Evaluate the mode of X.

(3)

(Total 11 marks)

 


 

8.       The cumulative frequency curve below indicates the amount of time 250 students spend eating lunch.

(a)     Estimate the number of students who spend between 20 and 40 minutes eating lunch.

(b)     If 20% of the students spend more than x minutes eating lunch, estimate the value of x.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

9.       A continuous random variable, X, has probability density function

          f (x) = sin x,   0 £ x £ .

          Find the median of X.

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 

 

10.     A fair six-sided die, with sides numbered 1, 1, 2, 3, 4, 5 is thrown. Find the mean and variance of the score.

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 


 

11.     A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.


 

(a)     Write down the number of students who scored 40 marks or less on the test.

(b)     The middle 50% of test results lie between marks a and b, where a < b. Find a and b.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

(Total 6 marks)

 


 

12.     The table below shows the probability distribution of a discrete random variable X.

x

0

1

2

3

P(X = x)

0.2

a

b

0.25

(a)     Given that E(X) = 1.55, find the value of a and of b.

(b)     Calculate Var(X).

(Total 6 marks)

 


 

13.     A random sample drawn from a large population contains the following data

                    6.2, 7.8, 12.1, 9.7, 5.2, 14.8, 16.2, 3.7.

Calculate an unbiased estimate of

(a)     the population mean;

(b)     the population variance.

(Total 6 marks)

 


 

14.     The following is the cumulative frequency diagram for the heights of 30 plants given in centimetres.

(a)     Use the diagram to estimate the median height.

..............................................................................................................................................

..............................................................................................................................................

 

(b)     Complete the following frequency table.

Height (h)

Frequency

0 £ h < 5

4

5 £ h < 10

9

10 £ h < 15

 

15 £ h < 20

 

20 £ h < 25

 


 

(c)     Hence estimate the mean height.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

(Total 6 marks)

 


 

15.     In a sample of 50 boxes of light bulbs, the number of defective light bulbs per box is shown below.

Number of defective light bulbs per box

0

1

2

3

4

5

6

Number of boxes

7

3

15

11

6

5

3

(a)     Calculate the median number of defective light bulbs per box.

(b)     Calculate the mean number of defective light bulbs per box.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

(Total 6 marks)

 


 

16.     A sample of discrete data is drawn from a population and given as

                66, 72, 65, 70, 69, 73, 65, 71, 75.

Find

(a)     the interquartile range;

(2)

(b)     an estimate for the mean of the population;

(2)

(c)     an unbiased estimate of the variance of the population.

(2)

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

(Total 6 marks)

 


 

17.     Consider the data set {k − 2, k, k +1, k + 4}, where kÎ.

(a)     Find the mean of this data set in terms of k.

(3)

Each number in the above data set is now decreased by 3.

(b)     Find the mean of this new data set in terms of k.

(2)

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

(Total 5 marks)

 

 

AssignmentWork Payment (USD)
Content