ST104a Statistics 1
ST104a Statistics 1
Chapter 2 – Solutions to Learning activities
Question 1
Work out the following:
(a) (2 + 4) × (3 + 7)
(b) 1/3 of 12 − 4 ÷ 2
(c) (1 + 4)/5 × (100 − 98).
If you find these difficult at all, go to Anthony and Biggs, or your old school
textbook and work on some more examples before you do anything else.
(a) Rules: Brackets take precedence so this is:
(6) × (10) = 60.
(b) Here the ‘of’ and ÷ take precedence and this is:
1
of 12 (= 4) minus 4 ÷ 2 (= 2)
3
i.e. 4 minus 2, i.e. +2.
(c) Brackets take precedence so we have:
5
×2
5
which is 1 × 2, which is 2.
You should find these really simple. If you are in any doubt at all, go over the rules again given
in Section 2.5 of the subject guide. Work through these examples again and (even better) go to
your old school arithmetic textbook and try some more!
Question 2
Work out the following (use a calculator where necessary):
√
(a) 16
(b) (0.07)2
√
(c) 0.49.
(a) 16 is 4 × 4 and −4 × −4, so the square root of 16 is ±4.
(b) (0.07)2 = 0.07 × 0.07 = 0.0049. Be careful with the decimal points!
(c) 0.49 is 0.7 × 0.7 and −0.7 × −0.7, so the square root of 0.49 is ±0.7.
Be careful that you understand the rules for placing decimal points. When you work on
estimation or tests of hypotheses of proportions you will need this.
1
,Chapter 2 – Solutions to Learning activities
Question 3
(a) What is 98% of 200?
(b) Give 17/25 as a percentage.
(c) What is 25% of 98/144?
(a) The answer is (98/100) × 200 which is 98 × 2, or 196.
(b) To get a percentage, multiply the given fraction by 100, i.e. we obtain:
17 17
× 100 = × 4 = 17 × 4 = 68%.
25 1
So 17/25 can be written as 68%.
(c) 25% of 98/144 is another way of writing (25/100) × (98/144) which is (1/4) × (49/72) or 49/288
which is 0.1701 to four decimal places.
Question 4
Give the absolute values for:
(a) |−8|
(b) |15 − 9|.
(a) This is, simply put, 8.
(b) This is the same as |6|, which is 6.
Question 5
(a) For which of 2, 3, 7 and 9 is x > 3?
(b) For which of 2, 3, 7 and 9 is x < 3?
(c) For which of 2, 3, 7 and 9 is x ≤ 3?
(d) For which of 2, 3, 7 and 9 is x2 ≥ 49?
(a) 7 and 9 are greater than 3.
(b) 2 is less than 3.
(c) 2 and 3 are less than or equal to 3.
(d) If x2 ≥ 49, then x ≥ 7. This is the case when x is 7 or 9.
Question 6
Given x1 = 3, x2 = 1, x3 = 4, x4 = 6 and x5 = 8, find:
5
P
(a) xi
i=1
4
x2i .
P
(b)
i=3
Given also that p1 = 1/4, p2 = 1/8, p3 = 1/8, p4 = 1/3 and p5 = 1/6, find:
5
P
(c) pi x i
i=1
2
, ST104a Statistics 1
5
pi x2i .
P
(d)
i=3
If you find these difficult, go back to an elementary textbook and do some more
work on this. It is most important that you deal with this before you embark on the
topic of descriptive statistics, such as means, in Chapter 4.
(a)
5
X
xi = x1 + x2 + x3 + x4 + x5
i=1
= 3+1+4+6+8
= 22.
(b)
4
X
x2i = x23 + x24
i=3
= 42 + 6 2
= 16 + 36
= 52.
(c)
5
X
pi xi = p1 x1 + p2 x2 + p3 x3 + p4 x4 + p5 x5
i=1
1 1 1 1 1
= ×3+ ×1+ ×4+ ×6+ ×8
4 8 8 3 6
3 1 1 1
= + + +2+1
4 8 2 3
18 + 3 + 12 + 0 + 8
= 3+
24
41
= 3+
24
17
= 4 or 4.7083, if you prefer to work in decimals.
24
(Note: do not forget the ‘times’ sign takes precedence over the ‘plus’. Check over Question 1 if in
any doubt!)
(d)
5
X
pi x2i = p3 x23 + p4 x24 + p5 x25
i=3
1 1 1
= × 42 + × 62 + × 82
8 3 6
1 1 1
= × 16 + × 36 + × 64
8 3 6
2
= 2 + 12 + 10
3
2
= 24 or 24.6667, if you prefer to work in decimals.
3
Check this carefully if you could not do it yourself first time. Note that i = 3 and i = 5 mark the
beginning and end, respectively, of the summation, as did i = 3 and i = 4 in (b).
3
, Chapter 2 – Solutions to Learning activities
Question 7
Sketch the following:
(a) y = x + 3
(b) y = 3x − 2.
You are going to need equations like this for all the material on regression in
Chapter 12.
(a) Here the straight line graph has a slope of 1 and the line cuts the y-axis at x = 3 (when x is zero,
y is 3).
y
3
-3 x
(b) Here the straight line still has a positive slope of 3 (y increases by 3 when x increases by 1) but
the line crosses the y-axis at a minus value (−2).
y
2/3 x
-2
Solutions prepared by Dr James Abdey.
4
ST104a Statistics 1
Chapter 2 – Solutions to Learning activities
Question 1
Work out the following:
(a) (2 + 4) × (3 + 7)
(b) 1/3 of 12 − 4 ÷ 2
(c) (1 + 4)/5 × (100 − 98).
If you find these difficult at all, go to Anthony and Biggs, or your old school
textbook and work on some more examples before you do anything else.
(a) Rules: Brackets take precedence so this is:
(6) × (10) = 60.
(b) Here the ‘of’ and ÷ take precedence and this is:
1
of 12 (= 4) minus 4 ÷ 2 (= 2)
3
i.e. 4 minus 2, i.e. +2.
(c) Brackets take precedence so we have:
5
×2
5
which is 1 × 2, which is 2.
You should find these really simple. If you are in any doubt at all, go over the rules again given
in Section 2.5 of the subject guide. Work through these examples again and (even better) go to
your old school arithmetic textbook and try some more!
Question 2
Work out the following (use a calculator where necessary):
√
(a) 16
(b) (0.07)2
√
(c) 0.49.
(a) 16 is 4 × 4 and −4 × −4, so the square root of 16 is ±4.
(b) (0.07)2 = 0.07 × 0.07 = 0.0049. Be careful with the decimal points!
(c) 0.49 is 0.7 × 0.7 and −0.7 × −0.7, so the square root of 0.49 is ±0.7.
Be careful that you understand the rules for placing decimal points. When you work on
estimation or tests of hypotheses of proportions you will need this.
1
,Chapter 2 – Solutions to Learning activities
Question 3
(a) What is 98% of 200?
(b) Give 17/25 as a percentage.
(c) What is 25% of 98/144?
(a) The answer is (98/100) × 200 which is 98 × 2, or 196.
(b) To get a percentage, multiply the given fraction by 100, i.e. we obtain:
17 17
× 100 = × 4 = 17 × 4 = 68%.
25 1
So 17/25 can be written as 68%.
(c) 25% of 98/144 is another way of writing (25/100) × (98/144) which is (1/4) × (49/72) or 49/288
which is 0.1701 to four decimal places.
Question 4
Give the absolute values for:
(a) |−8|
(b) |15 − 9|.
(a) This is, simply put, 8.
(b) This is the same as |6|, which is 6.
Question 5
(a) For which of 2, 3, 7 and 9 is x > 3?
(b) For which of 2, 3, 7 and 9 is x < 3?
(c) For which of 2, 3, 7 and 9 is x ≤ 3?
(d) For which of 2, 3, 7 and 9 is x2 ≥ 49?
(a) 7 and 9 are greater than 3.
(b) 2 is less than 3.
(c) 2 and 3 are less than or equal to 3.
(d) If x2 ≥ 49, then x ≥ 7. This is the case when x is 7 or 9.
Question 6
Given x1 = 3, x2 = 1, x3 = 4, x4 = 6 and x5 = 8, find:
5
P
(a) xi
i=1
4
x2i .
P
(b)
i=3
Given also that p1 = 1/4, p2 = 1/8, p3 = 1/8, p4 = 1/3 and p5 = 1/6, find:
5
P
(c) pi x i
i=1
2
, ST104a Statistics 1
5
pi x2i .
P
(d)
i=3
If you find these difficult, go back to an elementary textbook and do some more
work on this. It is most important that you deal with this before you embark on the
topic of descriptive statistics, such as means, in Chapter 4.
(a)
5
X
xi = x1 + x2 + x3 + x4 + x5
i=1
= 3+1+4+6+8
= 22.
(b)
4
X
x2i = x23 + x24
i=3
= 42 + 6 2
= 16 + 36
= 52.
(c)
5
X
pi xi = p1 x1 + p2 x2 + p3 x3 + p4 x4 + p5 x5
i=1
1 1 1 1 1
= ×3+ ×1+ ×4+ ×6+ ×8
4 8 8 3 6
3 1 1 1
= + + +2+1
4 8 2 3
18 + 3 + 12 + 0 + 8
= 3+
24
41
= 3+
24
17
= 4 or 4.7083, if you prefer to work in decimals.
24
(Note: do not forget the ‘times’ sign takes precedence over the ‘plus’. Check over Question 1 if in
any doubt!)
(d)
5
X
pi x2i = p3 x23 + p4 x24 + p5 x25
i=3
1 1 1
= × 42 + × 62 + × 82
8 3 6
1 1 1
= × 16 + × 36 + × 64
8 3 6
2
= 2 + 12 + 10
3
2
= 24 or 24.6667, if you prefer to work in decimals.
3
Check this carefully if you could not do it yourself first time. Note that i = 3 and i = 5 mark the
beginning and end, respectively, of the summation, as did i = 3 and i = 4 in (b).
3
, Chapter 2 – Solutions to Learning activities
Question 7
Sketch the following:
(a) y = x + 3
(b) y = 3x − 2.
You are going to need equations like this for all the material on regression in
Chapter 12.
(a) Here the straight line graph has a slope of 1 and the line cuts the y-axis at x = 3 (when x is zero,
y is 3).
y
3
-3 x
(b) Here the straight line still has a positive slope of 3 (y increases by 3 when x increases by 1) but
the line crosses the y-axis at a minus value (−2).
y
2/3 x
-2
Solutions prepared by Dr James Abdey.
4
