Download 13 Univariate Data

13. UNIVARIATE DATA
1.
Do customers spend more on petrol when petrol prices are low? In order to investigate this
question, a garage owner took two random samples of 100 customers being careful to
choose the same day of the week and during the same hours of the day. One sample was
taken when the petrol price was high and one when the petrol price was low. The amount
spent by each customer was rounded to the nearest 10 cents. A frequency table for the data
collected is given below:
Amount $
0.10 - 5.00
5.10 - 10.00
10.10-15.00
15.10-20.00
20.10-25.00
25.10-30.00
High
Low
10
8
12
9
12
12
32
25
25
30
9
16
The average amount spent when petrol was at the high price was $16.40.
(a) [2 marks] Is the average amount spent when petrol was at the low price more than
the average amount spent when petrol was at the high price? Justify your answer.
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(b) [1 mark] Calculate the standard deviation for the amount spent when petrol was at
the low price.
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(c) [2 marks] Calculate the median amount spent when petrol was at the high price.
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VET Mathematics
97
13. Univariate Data
2.
A class consists of 15 girls and 20 boys. The mean weight of the girls is 38 1/3 kg, whilst the
boys have a mean weight of 50kg with a standard deviation of 5kg. The whole class has a
mean weight of 45kg with a standard deviation of 9kg.
(a) [6 marks] Calculate the variance of the weight of the girls
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A weight in kilograms can be converted to a weight in pounds by multiplying by 2.2.
(b) [1 mark]
What is the variance of the weight of the girls if their weights are given in
pounds?
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3.
Consider the following data, which represent the number of people entering a fast food
outlet in 10 consecutive half-hour periods on a particular day.
30
20
40
30
46
20
14
36
47
17
One way of testing whether or not data could be symmetrically distributed is to compare
the mean and the median values. If the data are symmetrical then the mean and the median
will be the same.
Could the above data be symmetrically distributed? Justify your answer.
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VET Mathematics
98
13. Univariate Data
4.
The variable x has values
-1
2
5
-2
6
1
4
3
-3
5
with a mean x = 2.
2
(a) [2 marks] Calculate the variance s .
x
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(b) [5 marks] Find the EXACT values of the mean and variance for the following data
3517892.3239619
3517892.3239622
3517892.3239625
3517892.3239618
3517892.3239626
3517892.3239621
3517892.3239624
3517892.3239623
3517892.3239617
3517892.3239625
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VET Mathematics
99
13. Univariate Data
5.
The following screen from a graphics calculator shows two boxplots constructed from two
sets of height data.
Data Set 1 contains the heights of a random sample of 30 adult females and is given below.
Data Set 2, also given below, contains some summary data for the heights of a random
sample of 30 adult males.
All heights were measured to the nearest centimetre.
Data Set 1 :
154 155 156
164 164 164
170 170 170
Data Set 2 :
Minimum
169
158
164
170
160
165
170
Q1
176
160
165
173
160
167
173
160
167
175
Median
180
161
168
175
Q3
184
163
168
176
Maximum
191
(a) [3 marks] Find the median, first quartile and third quartile for Data Set 1.
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(b) [2 marks] Calculate the range for each data set.
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(c) [1 marks] Calculate the interquartile range for each data set.
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(d)[1 marks] Indicate on the boxplots which represents the male heights and which the
female heights.
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(e) [3 marks] With reference to the boxplots and any computed values, comment upon
any similarities and any differences in the distribution of the two data sets.
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VET Mathematics
100
13. Univariate Data
6.
For the following data set (given in ascending order)
a
6
7
7
the mean is 10.75;
b
c
11
the range is 13;
12
13
d
the median is 10;
17
18
the inter quartile range is 7.
Find the values of a, b, c and d.
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7.
A summary of a data set X is given below.
n  52
x  27.5
Q1 ( x)  13
sx  12.0
median  25
Q3 ( x)  35
A transformation y  100  2 x was applied to this data set. For the new data set Y find the
value of the
(a) [1 mark] mean,
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(b) [2 marks] first quartile,
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(c) [1 mark] median,
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(d)[2 marks] standard deviation.
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VET Mathematics
101
13. Univariate Data
8.
The set X of data illustrated on the box and whisker plot below has a mean of 7 and
a variance of 5.
4
5 6
8
12
Two different linear transformations of the original data are illustrated below.
Write down the mean and variance of each new set of data.
(a) [2 marks]
7
8 9
11
15
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(b) [2 marks]
8
10
12
16
24
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9.
In a certain year, Mrs Chan taught Calculus in one school and Applicable Mathematics in
another school. At the end of the year she summarised the percentage marks (rounded to
nearest whole number) obtained by the students in the two classes. This is given in the
frequency table below.
% mark range
Calculus
Applicable
0-19
1
2
20-39
3
2
40-59
3
5
60-79
9
11
80-99
5
7
Find the mean and standard deviation for the Calculus marks.
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VET Mathematics
102
13. Univariate Data
10.
A household kept the following record of local calls made from their home telephone for a
period of six months.
Month
Total number of calls
Total duration of calls (mins)
1
35
153
2
86
280
3
38
140
4
41
201
5
50
250
6
33
151
(a) For the six-month period, calculate
(i) [1 mark] The average duration of a local call,
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(ii) [2 marks] The standard deviation for the number of calls per month.
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(b) [3 marks] Suppose that local calls are charged at a rate of 5 cents per minute and the
telephone connection charge for a month is $5. Find the average telephone bill per month
for the household, assuming that only local calls are made.
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(c) [2 marks]
The number of local calls in a particular month can be thought of as
being unusually large if the number for the month is more than two standard deviations
larger than the average number per month for the six-month period. Justifying your
answer, list any months having an unusually large number of local calls, given that the
average number of calls per month for the six-month period is 47.17.
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VET Mathematics
103
13. Univariate Data
11.
The following data from the Australian Bureau of Statistics gives the number of
nights, in thousands, spent by international visitors to Australia in 1999 by State
(first six entries in table) or Territory.
NSW
38334
Vic
20494
Qld
24928
SA
4542
WA
12349
Tas
1694
NT
3044
ACT
2161
(a)
If there were 4.459510 million international visitors to Australia in 1999,
calculate the average number of nights spent in Australia per international
visitor.
[3]
(b)
For a randomly chosen international visitor night in 1999, calculate the
probability that it was
(i) spent in WA,
[2]
(i)
spent in one of the Territories (NT or ACT),
[2]
(ii)
not spent in NSW nor Qld.
[2]
VET Mathematics
104
13. Univariate Data
12.
A data set, X, consists of 40 data items, x i . Some summary values for the data set are given
below.
mean
16.5
median
15
minimum
10
standard deviation
2.5
maximum
27
1st quartile
13
3rd quartile
18
Unfortunately, when the data was recorded the largest data item was written down as 27
instead of 21.
For the correct data values
(a)
draw an appropriate box and whisker plot on the grid below,
[3]
5
10
(b)
15
20
25
30
write down the value of the mean,
[3]
40
(c)
give the value of the variance, if the original
x
i 1
2
i
 11140 .
[4]
VET Mathematics
105
13. Univariate Data