StatisticsIB MYP Maths Extended: Topic test

1. 1
   A ranger in Banff National Park wants to know how satisfied visitors are with the park. She stands at the main car park entrance on one Saturday morning and asks the first $50$ visitors who arrive.

   (a)

   Which sampling method is the ranger using?

   [1 mark]

   • AStratified sampling
   • BConvenience sampling
   • CSystematic sampling
   • DSimple random sampling

   (b)

   Why might the results be biased?

   [1 mark]

   • AA sample of $50$ is always too large to be reliable
   • BAsking visitors in person always causes bias
   • CParks cannot be surveyed on a Saturday
   • DVisitors who arrive on other days or at other entrances have no chance of being chosen

   (c)

   Describe two changes that would make the sample more representative.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A newspaper in Johannesburg prints a bar chart of its yearly sales. The bar for 2022 shows $10\,000$ copies and the bar for 2023 shows $10\,300$ copies. The vertical axis starts at $9\,800$ copies instead of $0$.

   (a)

   Which feature makes the increase in sales look larger than it really is?

   [1 mark]

   • AThe vertical axis starts at $9\,800$ instead of $0$
   • BThe bars are labelled with the years
   • CThe bars have equal widths
   • DThe chart has a title

   (b)

   By what percentage did sales actually increase from 2022 to 2023?

   [1 mark]

   • A$0.03\%$
   • B$30\%$
   • C$3\%$
   • D$103\%$

   (c)

   Measuring the bar heights from the start of the vertical axis, how many times taller does the 2023 bar look than the 2022 bar?

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bus company in Istanbul records the waiting times $t$ minutes of $60$ passengers. In the classes $0\le t<5$, $5\le t<10$, $10\le t<15$, $15\le t<20$ and $20\le t<25$ there are $6$, $18$, $24$, $9$ and $3$ passengers respectively.

   (a)

   Estimate the mean waiting time.

   [3 marks]

   (b)

   (i) State the modal class.
   (ii) Find the class that contains the median.
   (iii) The company claims that the average wait is under $10$ minutes. Comment on this claim.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An ecologist measures the heights $h$ metres of $80$ trees in a forest in Sweden. The cumulative frequencies are: $5$ trees with $h\le10$, $15$ with $h\le20$, $40$ with $h\le30$, $65$ with $h\le40$ and $80$ with $h\le50$.

   (a)

   (i) Estimate the median, the lower quartile and the upper quartile.
   (ii) Find the interquartile range.
   (iii) Estimate how many trees are taller than the upper quartile.

   [6 marks]

   (b)

   (i) Estimate the mean height of the trees in this forest.
   (ii) A second forest has a median height of $34$ m and an interquartile range of $9$ m. Compare the two forests and decide which has the more consistent tree heights.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A sleep researcher in Auckland asks $12$ teenagers for the hours $x$ they spend on screens each evening, which range from $0.5$ to $6$, and the hours $y$ they sleep that night. On a scatter graph she draws a line of best fit that passes through the points $(1,\ 9.2)$ and $(5,\ 6.8)$.

   (a)

   Which statement describes the relationship?

   [1 mark]

   • APositive correlation: sleep tends to be longer when screen time is longer
   • BNo correlation: screen time and sleep are unrelated
   • CNegative correlation: sleep tends to be shorter when screen time is longer
   • DNegative correlation: longer screen time causes shorter sleep

   (b)

   What is the gradient of the line of best fit?

   [1 mark]

   • A$-0.6$
   • B$0.6$
   • C$-2.4$
   • D$-1.67$

   (c)

   Use the line of best fit to predict the hours of sleep for a teenager with $4$ hours of screen time.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two cafés in Wellington, Luna and Sol, count their customers in each of five consecutive hours. Luna: $20,\ 22,\ 24,\ 26,\ 28$. Sol: $10,\ 18,\ 24,\ 30,\ 38$. Both sets have a mean of $24$. Use the population standard deviation $\sigma$ throughout.

   (a)

   Technology gives $\sigma=2.83$ for Luna and $\sigma=9.63$ for Sol. Which statement is correct?

   [1 mark]

   • ASol serves more customers on average, because its standard deviation is larger
   • BLuna's numbers are more spread out, because its standard deviation is smaller
   • CThe two cafés are identical, because their means are equal
   • DLuna's numbers of customers are more consistent, because its standard deviation is smaller

   (b)

   Sol removes its lowest and highest hours ($10$ and $38$) from its data. What happens to the mean and the standard deviation of the remaining three values?

   [1 mark]

   • AThe mean stays at $24$ and the standard deviation increases
   • BThe mean stays at $24$ and the standard deviation decreases
   • CBoth the mean and the standard deviation stay the same
   • DThe mean decreases and the standard deviation decreases

   (c)

   Luna opens a second branch whose hourly customers are exactly double Luna's: $40,\ 44,\ 48,\ 52,\ 56$. Find the standard deviation of the second branch.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A cycling club in Lyon records the lengths $d$ km of its members' rides in one month: $12$ rides with $0<d\le10$, $30$ rides with $10<d\le20$, $36$ rides with $20<d\le40$ and $24$ rides with $40<d\le80$. The data are shown in a histogram.

   (a)

   Calculate the frequency density of each class.

   [3 marks]

   (b)

   Estimate the number of rides longer than $50$ km, and state the assumption you make.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A swimming coach in Brisbane records the weekly training time $x$ hours and the 50 m freestyle time $y$ seconds of ten swimmers. The training times range from $4$ to $14$ hours. Technology gives the line of best fit $y=-0.85x+36.4$ and the correlation coefficient $r=-0.94$.

   (a)

   (i) Describe the strength and direction of the correlation.
   (ii) Interpret the gradient and the $y$-intercept of the line in this context.
   (iii) Use the line to predict the time of a swimmer who trains for $10$ hours a week.

   [6 marks]

   (b)

   (i) Use the line to predict the time of a swimmer who trains for $30$ hours a week.
   (ii) Comment on the reliability of this prediction.
   (iii) A swimmer who trains for $9$ hours a week swims $29.5$ seconds. Compare this time with the prediction from the line.

   [6 marks]

   Total for question 8: 12 marks

End of questions