Statistics and probabilityIB Maths: Applications and Interpretation SL: Topic test

1. 1
   The number of parcels delivered per day by a courier is recorded over several months. The lower quartile is $14$ and the upper quartile is $26$.

   (a)

   Find the interquartile range.

   [1 mark]

   • A$40$
   • B$12$
   • C$20$
   • D$6$

   (b)

   Which of the following values would be classed as an outlier, using the rule that an outlier is more than $1.5\times\text{IQR}$ from the nearest quartile?

   [1 mark]

   • A$38$
   • B$43$
   • C$45$
   • D$2$

   (c)

   On one day the courier delivered $45$ parcels because a rival company had closed for the day. A manager suggests deleting this value. Explain whether the manager should delete it.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In a survey of $400$ commuters in Dubai, $148$ said that they usually travel to work by Metro.

   (a)

   Find an estimate of the probability that a commuter chosen at random usually travels by Metro.

   [1 mark]

   • A$0.37$
   • B$0.63$
   • C$0.148$
   • D$2.70$

   (b)

   There are $1500$ commuters in a company. Estimate the number who usually travel by Metro.

   [1 mark]

   • A$148$
   • B$945$
   • C$4054$
   • D$555$

   (c)

   Find the probability that a commuter chosen at random does not usually travel by Metro.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The length, $L$ cm, of fish of a certain species caught in a lake is normally distributed with mean $34.0$ cm and standard deviation $4.5$ cm.

   (a)

   Find the probability that a fish chosen at random has a length between $30$ cm and $40$ cm. Hence estimate how many of $500$ fish caught are between $30$ cm and $40$ cm in length. Use your GDC.

   [3 marks]

   (b)

   The longest $15\%$ of the fish are sold as premium fish. Find the least length of a premium fish. Premium fish sell for $12$ AED each and all other fish sell for $7$ AED each. Find the expected income from one fish chosen at random.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A seed supplier states that $85\%$ of its sunflower seeds germinate, independently of each other. A packet contains $20$ seeds. Let $X$ be the number of seeds that germinate in a packet.

   (a)

   Assuming that the supplier's statement is correct, find (i) $E(X)$, (ii) $\text{Var}(X)$, (iii) $P(X=17)$, (iv) $P(X\ge18)$. Use your GDC where appropriate.

   [6 marks]

   (b)

   The supplier sells each packet for $3$ AED. If fewer than $16$ seeds in a packet germinate, the supplier refunds $5$ AED to the customer. Let $G$ be the supplier's gain, in AED, from one packet. (i) Find $P(X\le15)$. (ii) Write down the probability distribution of $G$. (iii) Find $E(G)$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The ages, $a$ years, of $60$ visitors to a museum are summarised in classes of equal width. $0\le a<10$: $6$ visitors; $10\le a<20$: $12$; $20\le a<30$: $18$; $30\le a<40$: $15$; $40\le a<50$: $9$.

   (a)

   Write down the modal class.

   [1 mark]

   • A$0\le a<10$
   • B$10\le a<20$
   • C$20\le a<30$
   • D$30\le a<40$

   (b)

   Find an estimate for the mean age of the visitors.

   [1 mark]

   • A$26.5$ years
   • B$25$ years
   • C$1590$ years
   • D$21.5$ years

   (c)

   Find the percentage of visitors who are aged $40$ or over.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A coach records, for $10$ runners, the weekly training distance $x$ km (from $15$ to $55$) and the time $y$ minutes taken to run a $10$ km race. The regression line of $y$ on $x$ is $y=78.4-0.62x$. Pearson's product-moment correlation coefficient is $r=-0.91$ and Spearman's rank correlation coefficient is $r_s=-0.95$.

   (a)

   Which of the following is the correct interpretation of the gradient of the regression line?

   [1 mark]

   • AFor every extra kilometre trained each week, the race time increases by $0.62$ minutes.
   • BA runner who trains for $0$ km each week has a race time of $0.62$ minutes.
   • CThe race time falls by $62\%$ for every extra kilometre trained each week.
   • DOn average, each extra kilometre trained each week is associated with a race time $0.62$ minutes shorter.

   (b)

   Which of the following statements about $r_s=-0.95$ is correct?

   [1 mark]

   • A$r_s$ can only be used when the relationship between $x$ and $y$ is linear.
   • BThere is a strong negative monotonic relationship between $x$ and $y$.
   • C$95\%$ of the runners ran faster when they trained more.
   • D$r_s$ is more sensitive to outliers than Pearson's coefficient $r$.

   (c)

   Use the regression line to predict the time taken by a runner who trains $40$ km each week.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A university surveys $200$ students about where they mainly study: in the library, at home or in a café. Of the $100$ first-year students, $28$ choose the library, $46$ choose home and $26$ choose a café. Of the $100$ second-year students, $42$ choose the library, $38$ choose home and $20$ choose a café. A $\chi^2$ test for independence is carried out at the $5\%$ significance level to test whether the place of study is independent of the year of study.

   (a)

   State the null hypothesis and the alternative hypothesis, and write down the number of degrees of freedom.

   [3 marks]

   (b)

   Show that the expected frequency for second-year students who choose the library is $35$. Using the GDC, the value of the test statistic is $\chi^2=4.34$ and the critical value at the $5\%$ significance level is $5.99$. State the conclusion of the test, giving a reason.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A streaming service finds that $40\%$ of its subscribers are on the family plan. Of the family plan subscribers, $25\%$ cancel within a year; of the other subscribers, $35\%$ cancel within a year. Let $F$ be the event that a subscriber is on the family plan and $C$ the event that a subscriber cancels within a year.

   (a)

   (i) Find $P(C)$. (ii) Given that a randomly chosen subscriber cancelled within a year, find the probability that they were on the family plan.

   [6 marks]

   (b)

   (i) Show that $F$ and $C$ are not independent events. Three subscribers are chosen at random. Assume that their decisions to cancel are independent, and let $X$ be the number of them who cancel within a year. (ii) Find $P(X=2)$. (iii) Find $E(X)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions