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

1. 1
   A manager records the number of emails received on twelve working days. In order of size they are $14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 55$. The lower quartile is $18$ and the upper quartile is $27$.

   (a)

   A data item is an outlier if it is more than $1.5\times\text{IQR}$ above the upper quartile. Find the upper boundary for outliers.

   [1 mark]

   • A$36$
   • B$4.5$
   • C$40.5$
   • D$67.5$

   (b)

   Which statement about outliers is correct?

   [1 mark]

   • A$55$ is the only outlier
   • B$14$ and $55$ are outliers
   • CThere are no outliers
   • D$29$ and $55$ are outliers

   (c)

   The manager finds that the value $55$ was a typing error and should have been $35$. Find the mean number of emails after the correction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   At a cinema, $150$ customers are asked what they bought. $90$ bought popcorn ($P$), $70$ bought a drink ($D$) and $40$ bought both.

   (a)

   A customer is chosen at random. Find the probability that the customer bought popcorn or a drink or both.

   [1 mark]

   • A$0.267$
   • B$0.6$
   • C$0.2$
   • D$0.8$

   (b)

   A customer is chosen at random from those who bought a drink. Find the probability that the customer also bought popcorn.

   [1 mark]

   • A$0.444$
   • B$0.571$
   • C$0.267$
   • D$0.6$

   (c)

   Determine whether buying popcorn and buying a drink are independent events. Justify your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of downloads $D$ (in thousands) of a new app in week $w$ after its launch, for $w=1,2,3,4,5,6$, is $3.1, 4.6, 6.8, 10.3, 15.2, 22.9$ respectively.

   (a)

   Use your GDC to find an exponential model of the form $D=a\times b^{w}$, giving $a$ and $b$ to 3 significant figures. Use the model to predict the number of downloads in week $8$.

   [3 marks]

   (b)

   The GDC gives $R^2=0.9999$ for the exponential model and $R^2=0.923$ for the best linear model. (i) Interpret $R^2=0.923$. (ii) State, with a reason, which model fits the data better. (iii) Interpret the value of $b$ in context. (iv) Give a reason why the exponential model should not be used to predict downloads in week $30$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a school, a student's final score is $S=0.4P+0.6Q$, where $P$ and $Q$ are the marks on Paper 1 and Paper 2. $P$ and $Q$ are independent random variables with $E(P)=52$, $Var(P)=64$, $E(Q)=60$ and $Var(Q)=100$.

   (a)

   (i) Find $E(S)$. (ii) Find $Var(S)$. (iii) Find $Var(P-Q)$.

   [6 marks]

   (b)

   The distribution of $S$ is not known. A random sample of $36$ students is taken and $\bar S$ is the mean of their final scores. (i) Explain why $\bar S$ is approximately normally distributed, and state its mean and variance. (ii) Find $P(\bar S>58)$. (iii) A sample of $36$ students has $s_n^{2}=44.1$. Find an unbiased estimate of $Var(S)$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Typing errors in a manuscript occur independently at a uniform average rate of $0.5$ per page.

   (a)

   Let $X$ be the number of errors on one page. Find $P(X=2)$.

   [1 mark]

   • A$0.0758$
   • B$0.152$
   • C$0.184$
   • D$0.25$

   (b)

   Find the probability that there is at least one error in a section of two pages.

   [1 mark]

   • A$0.368$
   • B$0.393$
   • C$0.5$
   • D$0.632$

   (c)

   A student suggests using a binomial distribution for the number of errors on a page. Give two reasons why a Poisson distribution is more suitable.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The weather each day is sunny (S), cloudy (C) or rainy (R). The transition matrix $T$ gives the probability of moving from today's weather (columns, in the order S, C, R) to tomorrow's weather (rows, in the same order): $T=\begin{pmatrix}0.6&0.25&0.2\\0.3&0.5&0.4\\0.1&0.25&0.4\end{pmatrix}$.

   (a)

   Today is sunny. Find the probability that it is rainy the day after tomorrow.

   [1 mark]

   • A$0.1$
   • B$0.175$
   • C$0.01$
   • D$0.3$

   (b)

   The long-term probabilities of the three types of weather are given by an eigenvector of $T$. Which eigenvalue does it correspond to?

   [1 mark]

   • A$0$
   • B$0.5$
   • C$1$
   • D$-1$

   (c)

   Use your GDC to find the long-term probability that a day is sunny.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A bank surveys $250$ customers about how they prefer to bank. Of the $100$ customers under $35$, $34$ prefer the mobile app, $41$ prefer a branch and $25$ prefer the telephone. Of the $150$ customers aged $35$ or over, $26$ prefer the app, $52$ prefer a branch and $72$ prefer the telephone. A $\chi^2$ test for independence is carried out at the $5\%$ significance level.

   (a)

   (i) State the null and alternative hypotheses. (ii) Write down the degrees of freedom. (iii) Show that the expected frequency for customers under $35$ who prefer the app is $24$.

   [3 marks]

   (b)

   Use your GDC to find the $\chi^2$ statistic and the $p$-value. State the conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A pharmacy claims that $80\%$ of prescriptions are filled within $10$ minutes. An inspector checks $20$ randomly chosen prescriptions and counts the number $X$ filled within $10$ minutes. The inspector suspects the true proportion $p$ is less than $80\%$ and uses a $5\%$ significance level. Assume $X$ is binomial.

   (a)

   (i) State the null and alternative hypotheses. (ii) In the check, $13$ prescriptions are filled within $10$ minutes. Find the $p$-value and state the conclusion. (iii) Find the critical region for the test. (iv) Find the probability of a Type II error when the true value of $p$ is $0.6$.

   [6 marks]

   (b)

   A technician at the pharmacy repairs dispensing machines. The repair times are normally distributed. A random sample of $12$ repairs has mean $47.5$ minutes and $s_{n-1}=6.2$ minutes. (i) Use your GDC to find a $95\%$ confidence interval for the mean repair time. (ii) Interpret your interval in context. (iii) The technician claims the mean repair time is $45$ minutes. Comment on this claim. (iv) State why a $t$-distribution is used.

   [6 marks]

   Total for question 8: 12 marks

End of questions