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Summarising numerical data: the median, range, IQR and box plots
Questions and Answers
1.
A sample of 21 travellers was asked the number of nights they spent away from home last year. Their responses are summarised in the following stem-and-leaf plot.0 | 1 1 2 2 3 4 5 5 6 6 7 91 | 0 2 4 4 62 | 2 2 33 |4 | 4The median number of nights away from home is:
A.
6
B.
6.5
C.
7
D.
8
E.
9
Correct Answer C. 7
Explanation The stem-and-leaf plot shows the number of nights away from home for the 21 travellers. The numbers in the plot are arranged in ascending order. The median is the middle value in a set of data. In this case, there are 21 data points, so the median would be the 11th value. Looking at the plot, the 11th value is 7, so the median number of nights away from home is 7.
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2.
A sample of 21 travellers was asked the number of nights they spent away from home last year. Their responses are summarised in the following stem-and-leaf plot.0 | 1 1 2 2 3 4 5 5 6 6 7 91 | 0 2 4 4 62 | 2 2 33 |4 | 4The interquartile range (IQR) of nights away from home is:
A.
3.5
B.
11.5
C.
15
D.
22
E.
43
Correct Answer B. 11.5
Explanation The interquartile range (IQR) is a measure of the spread or variability of a dataset. It is calculated by subtracting the value of the first quartile (Q1) from the value of the third quartile (Q3). In this case, the first quartile is 2 and the third quartile is 13.5. Therefore, the IQR is 13.5 - 2 = 11.5.
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3.
The amount of money (in dollars) spent by each of 15 students on leisure activities in the past week is as follows:10 30 30 30 35 38 45 45 47 49 50 51 52 58 100The 5 figure summary for this data is:
A.
Minimum = 10, Q1 = 30, Median = 45, Q3 = 52, Maximum = 100
B.
Minimum = 10, Q1 = 30, Median = 46, Q3 = 51, Maximum = 100
C.
Minimum = 10, Q1 = 30, Median = 45, Q3 = 51.5, Maximum = 100
D.
Minimum = 10, Q1 = 30, Median = 45, Q3 = 51, Maximum = 100
Correct Answer D. Minimum = 10, Q1 = 30, Median = 45, Q3 = 51, Maximum = 100
Explanation The 5 figure summary provides a summary of the distribution of the data. The minimum value is the smallest value in the dataset, which is 10. The first quartile (Q1) represents the 25th percentile of the data, and it is 30. The median represents the middle value of the dataset, and it is 45. The third quartile (Q3) represents the 75th percentile of the data, and it is 51. Lastly, the maximum value is the largest value in the dataset, which is 100. This summary provides information about the spread and central tendency of the data.
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4.
The amount of money (in dollars) spent by each of 15 students on leisure activities in the past week is as follows:10 30 30 30 35 38 45 45 47 49 50 51 52 58 100which of the following statements is true?
A.
50% of students spent between 30 and 51 dollars
B.
25% of students spent less than $35
C.
25% of students spent more than $50
D.
75% of students spent less than $50
Correct Answer A. 50% of students spent between 30 and 51 dollars
Explanation To find the statement that is true, we need to analyze the given data. There are a total of 15 students. The amount of money spent by each student is arranged in ascending order. The 50% mark would be the median, which is the middle value when the data is arranged in ascending order. In this case, the median is between 30 and 51 dollars, as it falls between the 7th and 8th values. Therefore, the statement "50% of students spent between 30 and 51 dollars" is true.
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5.
The amount of money (in dollars) spent by each of 15 students on leisure activities in the past week is as follows:10 30 30 30 35 38 45 45 47 49 50 51 52 58 100The data value of $100 is an outlier.
A.
True
B.
False
Correct Answer A. True
Explanation The given answer, True, is correct because an outlier is a data point that is significantly different from other data points in a dataset. In this case, the amount of money spent by each student ranges from 10 to 100 dollars. However, the value of $100 is much higher than the other values, making it an outlier.
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6.
Two hundred people were asked about their attitude to compulsory voting (support, no opinion, do not support) and their age (in years).The variables Attitude to compulsory voting and Age are:
A.
Both categorical variables
B.
Both numerical variables
C.
A categorical and a numerical variable respectively
D.
A numerical and a categorical variable respectively
E.
Neither numerical nor categorical variables
Correct Answer C. A categorical and a numerical variable respectively
Explanation The variables "Attitude to compulsory voting" and "Age" are a categorical and a numerical variable respectively. "Attitude to compulsory voting" is a categorical variable because it represents different attitudes (support, no opinion, do not support) which are distinct categories. "Age" is a numerical variable because it represents a quantitative measurement of a person's age in years.
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7.
Two hundred people were asked about their attitude to compulsory voting (support, no opinion, do not support) and their age (in years).The most appropriate way to graphically display the information about Age is to use a:
A.
Dot plot
B.
Bar chart
C.
Segmented bar chart
D.
Histogram
E.
Back-to-back stem plot
Correct Answer D. Histogram
Explanation A histogram is the most appropriate way to graphically display the information about age in this scenario. A histogram is a visual representation of data that uses bars to represent the frequency or count of data points within specific intervals or bins. In this case, the age data can be grouped into intervals (e.g., 0-10, 11-20, 21-30, etc.) and the height of each bar in the histogram would represent the number of people within that age range. This allows for a clear visualization of the distribution of ages among the surveyed individuals.
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8.
Which of the following statements is correct.
A.
The above box plot has a symmetric distribution.
B.
The above box plot has a negatively skewed distribution.
C.
The above box plot has a positively skewed distribution.
D.
The above box plot has an increasingly positive distribution.
Correct Answer C. The above box plot has a positively skewed distribution.
Explanation The above box plot has a positively skewed distribution because the median line is closer to the lower quartile, indicating that the data is concentrated towards the higher values. Additionally, the whisker on the right side of the box plot is longer than the one on the left side, further indicating a skew towards higher values.
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9.
This distribution is approximately _________ .
Correct Answer symmetric
Explanation The given correct answer suggests that the distribution being referred to is approximately symmetric. This means that the data is evenly distributed on both sides of the central point, with similar frequencies of values occurring on either side. A symmetric distribution is often characterized by a bell-shaped curve, such as the normal distribution.
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