Independent events across days/trials

Calculate probabilities involving multiple independent observations from a normal distribution (e.g., probability all exceed a value, or exactly k exceed it).

30 questions

WJEC Further Unit 5 Specimen Q1
  1. Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm { N } \left( 32,4 ^ { 2 } \right)\). You may assume that the times taken to complete the crossword on successive days are independent.
    1. (i) Find the upper quartile of \(X\) and explain its meaning in context.
      (ii) Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes.
    2. Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm { N } \left( 18,2 ^ { 2 } \right)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword.
    3. A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows.
    $$\begin{array} { l l l l l l l l l l } 68 \cdot 1 & 70 \cdot 4 & 68 \cdot 6 & 67 \cdot 7 & 71 \cdot 3 & 67 \cdot 6 & 68 \cdot 9 & 70 \cdot 2 & 68 \cdot 4 & 69 \cdot 8 \end{array}$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\).
  2. Determine a 95\% confidence interval for \(\mu\).
  3. The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95 ?' Explain why the answer to this question is no and give a correct interpretation.
Edexcel S1 2022 January Q5
  1. Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4
    1. Find the probability that a particular value generated by the computer program is less than 37
    Jia changes the mean to \(m\) but leaves the standard deviation as 2.4
    The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
  2. Find the value of \(m\), giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
    The computer program then randomly generates 5 independent values from this normal distribution.
  3. Find the probability that at least one of these values is negative.
AQA S1 2010 January Q1
1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, \(X\) metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .
  1. Determine:
    1. \(\mathrm { P } ( X < 10.5 )\);
    2. \(\mathrm { P } ( 10.0 < X < 10.5 )\).
  2. A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.
Edexcel Paper 3 2018 June Q5
  1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.
Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.
  1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
  2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
  3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.
OCR Stats 1 2017 Specimen Q7
7
  1. The heights of English men aged 25 to 34 are normally distributed with mean 178 cm and standard deviation 8 cm .
    Three English men aged 25 to 34 are chosen at random.
    Find the probability that all three men have a height less than 194 cm .
  2. The diagram shows the distribution of heights of Scottish women aged 25 to 34 .
    \includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-08_559_1389_855_412} The distribution is approximately normal. Use the diagram in the Printed Answer Booklet to estimate the standard deviation of these heights, explaining your method.