IBDP Maths AA: Topic: SL 4.3: Measures of central tendency: IB style Questions SL Paper 2

Question: [Maximum mark: 4]

The number of hours spent exercising each week by a group of students is shown in the following table.

Exercising time
(in hours)
Number of
students
25
31
44
53
6x

The median is 4.5 hours.
(a) Find the value of x . 
(b) Find the standard deviation.

Answer/Explanation

Ans:

(a) EITHER
recognising that half the total frequency is 10 (may be seen in an ordered list or indicated on the frequency table)

OR
5+1+4 =3 + x
OR

\(\sum f = 20\)
THEN
x = 7

(b) METHOD 1
1.58429…
1.58
METHOD 2
EITHER

Question: [Maximum mark: 4]

The number of hours spent exercising each week by a group of students is shown in the following table.

Exercising time
(in hours)
Number of
students
25
31
44
53
6x

The median is 4.5 hours.
(a) Find the value of x . 
(b) Find the standard deviation.

Answer/Explanation

Ans:

(a) EITHER
recognising that half the total frequency is 10 (may be seen in an ordered list or indicated on the frequency table)

OR
5+1+4 =3 + x
OR

\(\sum f = 20\)
THEN
x = 7

(b) METHOD 1
1.58429…
1.58
METHOD 2
EITHER

Question: [Maximum mark: 4]

The number of hours spent exercising each week by a group of students is shown in the following table.

Exercising time
(in hours)
Number of
students
25
31
44
53
6x

The median is 4.5 hours.
(a) Find the value of x . 
(b) Find the standard deviation.

Answer/Explanation

Ans:

(a) EITHER
recognising that half the total frequency is 10 (may be seen in an ordered list or indicated on the frequency table)

OR
5+1+4 =3 + x
OR

\(\sum f = 20\)
THEN
x = 7

(b) METHOD 1
1.58429…
1.58
METHOD 2
EITHER

Question

Question

The weights in grams of 80 rats are shown in the following cumulative frequency diagram.

Write down the median weight of the rats.[1]

a(i).

Find the percentage of rats that weigh 70 grams or less.[3]

a(ii).

The same data is presented in the following table.

Weights \(w\) grams
\(0 \leqslant w \leqslant 30\)\(30 < w \leqslant 60\)\(60 < w \leqslant 90\)\(90 < w \leqslant 120\)
Frequency\(p\)\(45\)\(q\)\(5\)

Write down the value of \(p\).[2]

b(i).

The same data is presented in the following table.

Weights \(w\) grams
\(0 \leqslant w \leqslant 30\)\(30 < w \leqslant 60\)\(60 < w \leqslant 90\)\(90 < w \leqslant 120\)
Frequency\(p\)\(45\)\(q\)\(5\)

Find the value of \(q\).[2]

b(ii).

The same data is presented in the following table.

Weights \(w\) grams
\(0 \leqslant w \leqslant 30\)\(30 < w \leqslant 60\)\(60 < w \leqslant 90\)\(90 < w \leqslant 120\)
Frequency\(p\)\(45\)\(q\)\(5\)

Use the values from the table to estimate the mean and standard deviation of the weights.[3]

c.

Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).

Find the percentage of rats that weigh 70 grams or less.[2]

d.

Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).

A sample of five rats is chosen at random. Find the probability that at most three rats weigh 70 grams or less.[3]

e.
Answer/Explanation

Markscheme

50 (g)     A1     N1

[2 marks]

a(i).

65 rats weigh less than 70 grams     (A1)

attempt to find a percentage     (M1)

eg     \(\frac{{65}}{{80}},{\text{ }}\frac{{65}}{{80}} \times 100\)

81.25 (%) (exact), 81.3     A1     N3

[2 marks]

a(ii).

\(p = 10\)     A2     N2

[2 marks]

b(i).

subtracting to find \(q\)     (M1)

eg     \(75 – 45 – 10\)

\(q = 20\)     A1     N2

[2 marks]

b(ii).

evidence of mid-interval values     (M1)

eg     \(15, 45, 75, 105\)

\(\overline x  = 52.5\)   (exact), \(\sigma  = 22.5\)   (exact)     A1A1     N3

[3 marks]

c.

0.781650

78.2   (%)     A2     N2

[2 marks]

d.

recognize binomial probability     (M1)

eg     \(X \sim {\text{B}}(n,{\text{ }}p)\), \(\left( \begin{array}{c}5\\r\end{array} \right)\) \( \times {0.782^r} \times {0.218^{5 – r}}\)

valid approach     (M1)

eg     \({\text{P}}(X \leqslant 3)\)

\(0.30067\)

\(0.301\)     A1     N2

[3 marks]

e.

Question

The following table gives the examination grades for 120 students.


Find the value of

(i)     p ;

(ii)    q .

[4]
a(i) and (ii).

Find the mean grade.

[2]
b.

Write down the standard deviation.

[1]
c.
Answer/Explanation

Markscheme

(a) (i) evidence of appropriate approach     (M1)

e.g. \(9 + 25 + 35\) , \(34 + 35\)

\(p = 69\)     A1     N2

(ii) evidence of valid approach     (M1)

e.g. \(109 – \) their value of p, \(120 – (9 + 25 + 35 + 11)\)

\(q = 40\)     A1     N2

[4 marks]

a(i) and (ii).

evidence of appropriate approach     (M1)

e.g. substituting into \(\frac{{\sum {fx} }}{n}\), division by 120

mean \(= 3.16\)     A1     N2

[2 marks]

b.

1.09     A1     N1

[1 mark]

c.
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