4.14 Linear combinations of random variables and unbiased estimatesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   On Saturdays the number of customers at a café is the random variable $X$ with mean $120$ and variance $64$. On Sundays the number of customers is the random variable $Y$ with mean $90$ and variance $36$. $X$ and $Y$ are independent. On a Saturday the café's income is $I=15X+200$ AED.

   (a)

   Find $E(X+Y)$.

   [1 mark]

   • A$30$
   • B$105$
   • C$210$
   • D$10\,800$

   (b)

   Find $Var(X-Y)$.

   [1 mark]

   • A$100$
   • B$28$
   • C$10$
   • D$30$

   (c)

   Find the expected income on a Saturday.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The masses, in grams, of a random sample of six eggs from a farm are $58, 61, 63, 60, 57, 65$. Use your GDC where needed.

   (a)

   Find an unbiased estimate of the population mean mass of the eggs.

   [1 mark]

   • A$60.5$ g
   • B$61.0$ g
   • C$364$ g
   • D$60.7$ g

   (b)

   Find an unbiased estimate of the population variance of the masses.

   [1 mark]

   • A$7.56$ g$^2$
   • B$9.07$ g$^2$
   • C$2.75$ g
   • D$3.01$ g

   (c)

   The variance of the six masses, dividing by $6$, is $s_n^2=7.56$. Show how this gives the unbiased estimate $9.07$ from part (b), and explain why this estimate is used rather than $7.56$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bakery sells a loaf with mass $L$ grams, where $E(L)=800$ and $Var(L)=25$, and rolls whose individual masses $R$ grams have $E(R)=60$ and $Var(R)=4$. The mass of the loaf and the masses of different rolls are all independent.

   (a)

   A bag contains one loaf and five rolls, with total mass $W=L+R_1+R_2+R_3+R_4+R_5$. Find $E(W)$ and $Var(W)$.

   [3 marks]

   (b)

   A second bag contains one loaf and one roll, with the mass of the roll counted five times: $V=L+5R$. Find $E(V)$ and $Var(V)$. Explain why $Var(V)\ne Var(W)$ although $E(V)=E(W)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A courier company records the number of parcels $X$ delivered in one day by each driver in a random sample of $40$ drivers. The number of drivers who delivered $8, 9, 10, 11, 12, 13$ parcels was $3, 7, 12, 10, 6, 2$ respectively. Use your GDC where needed.

   (a)

   (i) Find an unbiased estimate of the population mean number of parcels.
   (ii) Find an unbiased estimate of the population variance.
   (iii) State what is meant by "unbiased" in this context.

   [6 marks]

   (b)

   Each driver is paid $6$ AED per parcel plus a fixed $40$ AED per day, so the payment is $P=6X+40$. Use your estimates from (a) as the population mean $\mu$ and variance $\sigma^2$.
   (i) Find $E(P)$.
   (ii) Two drivers deliver $X_1$ and $X_2$ parcels independently. Find $E(X_1+X_2)$ and $Var(X_1+X_2)$.
   (iii) Explain why $Var(X_1+X_2)$ is not equal to $Var(2X)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions