4.15 Central limit theoremIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The mass $X$ kg of a bag of flour is normally distributed with mean $1.00$ kg and standard deviation $0.04$ kg. A random sample of $9$ bags is taken, with masses independent of each other, and $\bar{X}$ is the mean mass of the sample.

   (a)

   Which of the following is the distribution of $\bar{X}$?

   [1 mark]

   • A$N\left(1,\ 0.04^2\right)$
   • B$N\left(9,\ \frac{0.04^2}{9}\right)$
   • C$N\left(1,\ \frac{0.04^2}{9}\right)$
   • D$N\left(1,\ \frac{0.04^2}{3}\right)$

   (b)

   Use your GDC to find $\mathrm{P}(\bar{X}>1.02)$.

   [1 mark]

   • A$0.309$
   • B$0.0668$
   • C$0.933$
   • D$0.134$

   (c)

   Explain why $\bar{X}$ is exactly normally distributed here, even though the sample size is only $9$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass of an adult passenger, $X$ kg, is normally distributed with mean $78$ kg and standard deviation $11$ kg. Passenger masses are independent of each other.

   (a)

   Six passengers are chosen at random. The total mass of the six passengers is $T$ kg. Which of the following is the distribution of $T$?

   [1 mark]

   • A$N(468,\ 726)$
   • B$N(468,\ 4356)$
   • C$N(78,\ 121)$
   • D$N(468,\ 66)$

   (b)

   Two passengers are chosen at random, with masses $X_1$ and $X_2$. Which of the following is the distribution of $X_1-X_2$?

   [1 mark]

   • A$N(156,\ 242)$
   • B$N(0,\ 0)$
   • C$N(0,\ 121)$
   • D$N(0,\ 242)$

   (c)

   A lift is overloaded if the total mass of six passengers exceeds $500$ kg. Use your GDC to find the probability that the lift is overloaded.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The waiting time $T$ minutes of a caller at a call centre has mean $6.0$ minutes and standard deviation $4.5$ minutes. The distribution of $T$ is positively skewed, so it is not normal. A random sample of $50$ calls is taken and $\bar{T}$ is the mean waiting time of the sample.

   (a)

   State the approximate distribution of $\bar{T}$, giving the reason why the approximation is valid and its parameters.

   [3 marks]

   (b)

   (i) Use your GDC to find the probability that the mean waiting time of the sample exceeds $7$ minutes.
   (ii) Explain why the same method cannot be used to find the probability that a single caller waits for more than $7$ minutes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bakery bakes loaves whose mass $L$ grams is normally distributed with mean $800$ g and standard deviation $15$ g. Each loaf is put in a paper bag whose mass $B$ grams is normally distributed with mean $20$ g and standard deviation $3$ g. All masses are independent.

   (a)

   (i) Write down the distribution of the mass $P=L+B$ of a loaf in its bag.
   (ii) Find the probability that a bagged loaf has mass less than $800$ g.
   (iii) Four bagged loaves are chosen at random. Find the probability that their total mass exceeds $3300$ g.
   Use your GDC where helpful.

   [6 marks]

   (b)

   An inspector weighs $36$ unbagged loaves chosen at random and finds their mean mass $\bar{L}$.
   (i) Write down the distribution of $\bar{L}$.
   (ii) Find $\mathrm{P}(\bar{L}<795)$.
   (iii) Use your GDC to find the smallest sample size $n$ for which $\mathrm{P}(\bar{L}<797)<0.01$.

   [6 marks]

   Total for question 4: 12 marks

End of questions