smc-644-10-Dot Plots

Data Analysis, GEN1 2023 VCAA 1-2 MC

The dot plot below shows the times, in seconds, of 40 runners in the qualifying heats of their 800 m club championship.

Question 1

The median time, in seconds, of these runners is

135.5
136
136.5
137
137

Question 2

The shape of this distribution is best described as

positively skewed with one or more possible outliers.
positively skewed with no outliers.
approximately symmetric with one or more possible outliers.
approximately symmetric with no outliers.
negatively skewed with one or more possible outliers.

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\(\text{Question 1:}\ B\)

\(\text{Question 2:}\ A\)

Show Worked Solution

\(\text{Question 1}\)

\(\text{40 data points}\ \Rightarrow \ \text{Median = average of 20th and 21st data points}\)

\(\text{Median}\ = \dfrac{136 + 136}{2} = 136\)

\(\Rightarrow B\)

\(\text{Question 2}\)

\(\text{Distribution is positive skewed (tail stretches to the right)} \)

\(\text{Q}_1 = \dfrac{135+135}{2} = 135\)

\(\text{Q}_3 = \dfrac{138+138}{2} = 138\)

\(\text{IQR} = 138-135=3 \)

\(\text{Outlier (upper fence)}\ = 138+ 1.5 \times 3 = 142.5\)

\(\Rightarrow A\)

CORE, FUR1 2018 VCAA 1-2 MC

The dot plot below displays the difference in travel time between the morning peak and the evening peak travel times for the same journey on 25 days.

Part 1

The percentage of days when there was five minutes difference in travel time between the morning peak and the evening peak travel times is

Part 2

The median difference in travel time is

3.0 minutes.
3.5 minutes.
4.0 minutes.
4.5 minutes.
5.0 minutes.

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`text(Part 1:)\ C`

`text(Part 2:)\ A`

Show Worked Solution

`text(Part 1)`

`text(Percentage)`	`= text(days with 5 m difference)/text(total days) xx 100`
	`= 5/25 xx 100`
	`= 20\ text(%)`

`=> C`

`text(Part 2)`

`text(Median)`	`= 13text(th data point)`
	`= 3\ text(minutes)`

`=> A`

CORE, FUR2 2016 VCAA 1

The dot plot below shows the distribution of daily rainfall, in millimetres, at a weather station for 30 days in September.

Write down the
1. range (1 mark)
2. median. (1 mark)
Circle the data point on the dot plot above that corresponds to the third quartile `(Q_3).` (1 mark)
Write down the
1. the number of days on which no rainfall was recorded (1 mark)
2. the percentage of days on which the daily rainfall exceeded 12 mm. (1 mark)
Use the grid below to construct a histogram that displays the distribution of daily rainfall for the month of September. Use interval widths of two with the first interval starting at 0. (2 marks)

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1. `17.8\ text(mm)`
2. `0`
`text(See Worked Solutions)`
1. `16\ text(days)`
2. `10\text(%)`
`text(See Worked Solutions)`

Show Worked Solution

a.i.	`text(Range)`	`=\ text(High) – text(Low)`
		`= 17.8 – 0`
		`= 17.8\ text(mm)`

a.ii. `text(30 data points)`

`text(Median)`	`= text{15th + 16th}/2`
	`= 0`

c.i. `16\ text(days)`

c.ii.	`text(Percentage)`	`= 3/30 xx 100`
		`= 10\ text(%)`

CORE, FUR2 2009 VCAA 1

Table 1 shows the number of rainy days recorded in a high rainfall area for each month during 2008.

The dot plot below displays the distribution of the number of rainy days for the 12 months of 2008.

Circle the dot on the dot plot that represents the number of rainy days in April 2008. (1 mark)
For the year 2008, determine
1. the median number of rainy days per month (1 mark)
2. the percentage of months that have more than 10 rainy days. Write your answer correct to the nearest per cent. (1 mark)

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1. `15.5`
2. `text(92%)`

Show Worked Solution

b.i.	`text(Median)`	`= text{(6th + 7th)}/2`
		`=(15+16)/2`
		`=15.5`

b.ii. `text(Months with more than 10 rainy days)`

`=11/12 xx text(100%)`

`=91.66…`

`=92text(%)\ \ text{(nearest %)}`

CORE, FUR2 2012 VCAA 1

The dot plot below displays the maximum daily temperature (in °C) recorded at a weather station on each of the 30 days in November 2011.

From this dot plot, determine
1. the median maximum daily temperature, correct to the nearest degree (1 mark)
2. the percentage of days on which the maximum temperature was less than 16 °C.
  
  Write your answer, correct to one decimal place. (1 mark)

Records show that the minimum daily temperature for November at this weather station is approximately normally distributed with a mean of 9.5 °C and a standard deviation of 2.25 °C.

Determine the percentage of days in November that are expected to have a minimum daily temperature less than 14 °C at this weather station.

Write your answer, correct to one decimal place. (1 mark)

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1. `20 text(°C)`
2. `23.3text(%)`
`97.5text(%)`

Show Worked Solution

a.i. `text(The median is the average of the 15th and 16th)`

`\ \ text{data points (30 data points in total).}`.

`:.\ text(Median = 20 °C)`

a.ii. `text{Percentage of days less than 16 °C}`

`= 7/30 xx 100text(%)`

`=23.333…`

`=23.3text{% (to 1 d.p.)}`

b.	`z text{-score (14)}`	`=(x-barx)/s`
		`=(14-9.5)/2.25`
		`=2`

`text(Percentage of days with a minimum below 14 °C)`

`= 95text(%) + 2.5text(%)`

`= 97.5text(%)`

CORE, FUR2 2013 VCAA 2

The development index for each country is a whole number between 0 and 100.

The dot plot below displays the values of the development index for each of the 28 countries that has a high development index.

Using the information in the dot plot, determine each of the following.

(1 mark)
Write down an appropriate calculation and use it to explain why the country with a development index of 70 is an outlier for this group of countries. (2 marks)

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`text(Mode = 78, Range = 9)`
`text(See Worked Solutions)`

Show Worked Solution

a. `text(Mode = 78)`

`text(Range = 79 − 70 = 9)`

b. `text(An outlier occurs if a data point is below)`

`Q_1 − 1.5 xx IQR`

`Q_1 = 75, \ \ Q_3 = 78, and IQR = 78-75=3`

`:. Q_1 − 1.5 xx IQR`	`= 75 − 1.5 xx 3`
	`= 70.5`

`:. 70\ text{is an outlier (70 < 70.5)}`

CORE, FUR1 2015 VCAA 3 MC

The dot plot below displays the difference between female and male life expectancy, in years, for a sample of 20 countries.

The mean (`barx`) and standard deviation (`s`) for this data are

A. `text(mean)\ = 2.32`	`\ \ \ \ \ text(standard deviation)\ = 5.25`
B. `text(mean)\ = 2.38`	`\ \ \ \ \ text(standard deviation)\ = 5.25`
C. `text(mean)\ = 5.0`	`\ \ \ \ \ text(standard deviation)\ = 2.0`
D. `text(mean)\ = 5.25`	`\ \ \ \ \ text(standard deviation)\ = 2.32`
E. `text(mean)\ = 5.25`	`\ \ \ \ \ text(standard deviation)\ = 2.38`

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`E`

Show Worked Solution

`text(By calculator.)`

`=> E`

CORE, FUR1 2007 VCAA 1-2 MC

The dot plot below shows the distribution of the number of bedrooms in each of 21 apartments advertised for sale in a new high-rise apartment block.

Part 1

The mode of this distribution is

A. `1`

B. `2`

C. `3`

D. `7`

E. `8`

Part 2

The median of this distribution is

A. `1`

B. `2`

C. `3`

D. `4`

E. `5`

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`text (Part 1:)\ A`

`text (Part 2:)\ B`

Show Worked Solution

`text (Part 1)`

`rArr A`

`text (Part 2)`

`text(The median of 21 data points is the 11th value.)`

`:.\ text(Median) = 2`

`rArr B`

CORE, FUR1 2008 VCAA 5 MC

A sample of 14 people were asked to indicate the time (in hours) they had spent watching television on the previous night. The results are displayed in the dot plot below.

Correct to one decimal place, the mean and standard deviation of these times are respectively

A. `bar x=2.0\ \ \ \ \ s=1.5`

B. `bar x=2.1\ \ \ \ \ s=1.5`

C. `bar x=2.1\ \ \ \ \ s=1.6`

D. `bar x=2.6\ \ \ \ \ s=1.2`

E. `bar x=2.6\ \ \ \ \ s=1.3`

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`C`

Show Worked Solution

`text(Data points are:)`

`0,0,0,1,1,2,2,2,2,3,3,4,4,5`

`text(By calculator (using sample standard deviation))`

`bar x=2.1,\ \ s=1.6`

`=> C`