1 Variable Statistics IB Questions: Central Measures 1. The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 (a) Find the value of a. (b) Find the median of these numbers. (Total 4 marks) 2. The table shows the number of children in 50 families. Number of children Frequency Cumulative frequency 1 3 3 2 m 22 3 12 34 4 p q 5 5 48 6 2 50 T (a) Write down the value of T. (b) Find the values of m, p and q. (Total 4 marks) 7. David looked at a passage from a book. He recorded the number of words in each sentence as shown in the following frequency table. (a)
Class interval (number of words) Frequency f 1–5 16 6–10 28 11–15 26 16–20 14 21–25 10 26–30 3 31–35 1 36–40 0 41–45 2 Find the class interval in which the median lies. (b) Estimate, correct to the nearest whole number, the mean number of words in a sentence. (Total 4 marks) 1
9. An atlas gives the following information about the approximate population of some cities in the year 2000. The population of Nairobi has accidentally been left out. City Population in Millions Melbourne 3.2 Bangkok 7.2 Nairobi Paris 9.6 São Paulo 17.7 Tokyo 28.0 Seattle 2.1 The atlas tells us that the mean population for this group of cities is 10.01 million. (a) Calculate the population of Nairobi. (b) Which city has the median population value? (Total 8 marks) 10. Peter has marked 80 exam scripts. He has calculated the mean mark for the scripts to be 62.1. Maria has marked 60 scripts with a mean mark of 56.8. (a) Peter discovers an error in his marking. He gives two extra marks each to eleven of the scripts. Calculate the new value of the mean for Peter’s scripts. (b) After the corrections have been made and the marks changed, Peter and Maria put all their scripts together. Calculate the value of the mean for all the scripts. (Total 8 marks) 18. For the set of {8, 4, 2, 10, 2, 5, 9, 12, 2, 6} (a) calculate the mean; (b) find the mode; (c) find the median. (Total 4 marks) 20. In the following ordered data, the mean is 6 and the median is 5. 2, b, 3, a, 6, 9, 10, 12 Find each of the following (a) the value of a; (b) the value of b. (Total 8 marks) 2
15. The numbers of games played in each set of a tennis tournament were 9, 7, 8, 11, 9, 6, 10, 8, 12, 6, 8, 13, 7, 9, 10, 9, 10, 11, 12, 8, 7, 13, 10, 7, 7. The raw data has been organized in the frequency table below. games frequency 6 2 7 5 8 n 9 4 10 4 11 2 12 2 13 2 (a) Write down the value of n. (b) Calculate the mean number of games played per set. (c) What percentage of the sets had more than 10 games? (d) What is the modal number of games? (Total 8 marks) 21. Twenty students are asked how many detentions they received during the previous week at school. The results are summarised in the frequency distribution table below. Number of detentions x Number of students f fx 0 6 1 3 2 10 3 1 Total 20 (a) What is the modal number of detentions received? (b) (i) Complete the table. (ii) Find the mean number of detentions received. (Total 4 marks) 3
1 Variable Statistics IB Questions: Central Measures -­‐ Answers 1.
(a)
(b)
4+3+ a +8+7+3+9+5+8+3
10
55 = 50 + a
5=a
5.5 =
3, 3, 3, 4, 5, 5, 7, 8, 8, 9
Median = 5
Note: Award (M1) for arranging scores in ascending or
descending order. Follow through with candidate’s a
(M1)
(A1)(C2)
(M1)
(A1)(C2)
[4]
2.
(a)
T = 50
(A1)
(b)
m = 19
(A1)
(c)
p=9
(A1)
(d)
q = 43
(A1)
[4]
7.
(a)
Interval 11–15
(A1)
(b)
Mid-intervals 3, 8, 13, 18 ...
Note: Award (M1) for all correct numbers.
(M1)
Σxf = 48 + 224 + 338 + ...
Note: Award (M1) for attempt to obtain sum.
(M1)
Mean = 13
(A1)
[4]
9.
(a)
(b)
3.2 + 7.2 + x + 9.6 + 17.7 + 28.0 + 2.1
= 10.01.
7
Hence 67.8 + x = 10.01 × 7 = 70.07.
x = 70.07 – 67.8 = 2.27 (accept 2.27 or 2.3 million).
µ=
Median is the middle value which is 7.2.
Bangkok.
Note: Award (M0)(A0)(A0) for Paris.
(M1)
(A1)(M1)
(M1)(A1)
(M1)(A1)
(A1)(C3)
[8]
4
10.
(a)
80 × 62.1 + 2 ×11 = 4990
Note: Award (M0)(A0) if 2 × 11 is subtracted and ft the
remainder of the question to answers of 61.825 (or 61.8) and
59.7 respectively.
4990
= 62.375 (or 62.4 to 3 s.f.)
80
(b)
4990 + 56.8 × 60 = 8398
8398
= 60.0 (3 s.f.)
140
Note: An answer of 60 (2 s.f.) with no working receives (G2) or
with working using 4990 receives (M1)(A1)(M1)(A0) AP,
however, if 80 × 62.4 is used then 60 is an exact answer and
can receive all the marks.
(M1)(A1)
(M1)(A1)
(M1)(A1)
(M1)(A1)
[8]
15.
(a)
n=4
(A2)(C2)
(b)
Mean number of games is 9.08 (accept 9).
Note: Award (M1) for indicating a sum of games times
frequency (possibly curtailed by dots) or for 227 seen.
(M1)(A1)
(c)
6 100
= 24%
×
25 1
(M1)(A1)
Note: Award (M1)(A0) if 6 is replaced by 10. No other
alternative.
(d)
Modal number of games is 7.
(A2)(C2)
[8]
18.
(a)
Mean =
=6
60
10
(A1)(C1)
(b)
Mode = 2
(A1)(C1)
(c)
2, 2, 2, 4, 5, 6, 8, 9, 10, 12
↑
Median = 5 + 6
2
= 5.5
(M1)
(A1)(C2)
[4]
5
20.
(a)
(b)
a+6
=5
2
a + 6 = 10
a=4
(M1)(A1)
(A1)
(A1)(C4)
42 + a + b
=6
8
(M1)
42 + a + b = 48
a+b=6
4+b=6
b=2
(A1)
(A1)
(A1)(C4)
[8]
21.
(a)
Mode = 2
(b)
(i)
(A1)
x
f
fx
0
1
2
3
6
3
10
1
0
3
20
3
Total
20
26
(A1)
Note: Award (A1) for three or more correct bold entries.
(ii)
Mean =
26
20
(M1)
Note: Award (M1) for dividing fx total by 20.
= 1.3
(A1)
Mean = 1.3
(C2)
OR
[4]
6