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1. Given that $X\sim N(0,1)$, find:
  1. $\mathrm{P}(X < 0.5)$
  2. $\mathrm{P}(X > -0.1)$
2. Given that $X\sim N(1,0.5)$, find:
  1. $\mathrm{P}(X > 0 | X < 0.5)$
  2. $\mathrm{P}(X < 1.3 \cup X > 1.5)$
3. In the last $60$ weeks, the total number of hours of sunshine was $1500$. In $10$ of those weeks, the number of hours of sunshine exceeded $30$ hours. Assuming the number of hours of sunshine per week can be modelled by a normal distribution, suggest suitable parameters for the distribution.
4. Given that $X\sim N(5,10)$, find:
  1. $\mathrm{P}(6 < |X|)$
  2. $\mathrm{P}(X > 6 | 6 < |X|)$
5. IQ tests have a mean of $100$ and standard deviation of $20$. What score is needed to be in the top $1$% of the population?
6. Given that $X\sim N(0,1)$, find $x$ such that:
  1. $\mathrm{P}(X < x) = 0.6$
  2. $\mathrm{P}(|X| < x) = 0.6$
7. A student's maths test score is normally distributed with mean $60$ and standard deviation $7$.
  1. What is the probability the student scores between $55$ and $65$ in a test?
  2. The student sits $3$ maths tests one day. What is the probability that the student scores more than $75$ in exactly one of the tests?
  3. What assumptions did you have to make in answering the previous question? Are these likely to be valid?
8. The mean time a group of students can run a $100$m race is $16$ seconds, with a standard deviation of $1$ second.
  1. What is the probability that a randomly chosen student can run it in under $15$ seconds?
  2. What is the probability that a team of four students each run under $15$ seconds?
  3. Are the students more or less likely than this to take a total time of less than a minute if they run sequentially in a relay?
9. The heights of some students have a mean of $160$ cm, a median of $159$ cm, and a standard deviation of $5$ cm.
  1. Explain why a normal distribution could be a reasonable model for these data.
  2. What height should a student be to be in the tallest $10$% of students?
10. Given $X\sim N(8\sigma,\sigma^2)$, find $\mathrm{P}(X < 6\sigma)$
11. Given $X\sim N(\mu,1)$, find $\alpha$ such that $$\mathrm{P}(\mu - \alpha < X < \mu + \alpha) = 0.4$$
12. The time taken for students to eat lunch can be modelled by a normal distribution with mean $27$ minutes and standard deviation $3$ minutes. Two randomly selected students both took longer than $x$ minutes to eat lunch, and the probability of this happening is $0.0952$. Find $x$.
13. Given $X\sim N(\mu,\sigma^2)$, find $\mu$ and $\sigma$ when:
  1. $\mathrm{P}(X > 3) = 0.6$ and $\mathrm{P}(X < 1) = 0.2$
  2. $\mathrm{P}(X > 20) = 0.3$ and $\mathrm{P}(X \geqslant 30) = 0.15$
14. The length of time a phone battery lasts can be modelled by a normal distribution. Out of 100 phones, 15 lasted less than 5 hours and 30 lasted more than 8 hours. Estimate the mean and standard deviation.
15. It was found that 25% of test scores are 5, or more, lower than the average. Assuming test scores follow a normal distribution, estimate the standard deviation.
16. The amount of pocket money some students earn in a year are shown below.
  1. What features of the diagram suggest a normal distribution is appropriate to model the amount of money students earn?
  2. Estimate the mean and standard deviation.
17. Tom wishes to use a normal distribution to approximate $X$, where $X\sim B(200, 0.3)$. Find the parameters of the normal distribution Tom should use.
18. The heights of a group of school children are normally distributed with mean $150$ cm and variance $25$ cm2. A student can only go on a rollercoaster ride if they are over $140$ cm tall. Given that the student is allowed on the rollercoaster, what is the probability that they are over $150$ cm tall?
19. A student has to wait $10$ minutes, on average, for a bus. $4$ out of $5$ times, the student waits for over $9$ minutes. If the student waits for over $12$ minutes, they will be late for school.
  1. What is the probability the student is late for school on any given day?
  2. What is the probability the student is never late to school in one week?
20. Given that $X \sim N(\mu,\sigma^2)$, find the interquartile range in terms of $\sigma$.
21. The following data show the amount of time, $t$ minutes, students spend on homework on an average evening:
$t$Frequency
$0 < t \leqslant 10$$1$
$10 < t \leqslant 20$$4$
$20 < t \leqslant 30$$8$
$30 < t \leqslant 40$$18$
$40 < t \leqslant 50$$9$
$50 < t \leqslant 60$$5$
  1. What features of this data make a normal distribution a suitable model?
  2. Assuming a normal distribution is appropriate, estimate the mean and standard deviation of times students spend on homework.
  3. Three students are selected at random. What is the probability they all spend more than $40$ minutes on homework in an average evening?
22. On average, school children spend $3.2$ hours a day on their phones. $20$% of children spend more than $4.1$ hours on their phones. Assuming the time spent on phones can be modelled by a normal distribution:
  1. What is the standard deviation of this normal distribution?
  2. What is the probability that, out of a group of $10$ friends, fewer than $3$ spend more than $4$ hours on their phones?
  3. What assumptions did you have to make in the previous question, and are they likely to be valid?
23. At a winter fair, students have to guess the mass of a large bar of chocolate. $20$% of guesses are less than $800$ grams, and $10$% of guesses are more than $1.4$ kilograms. It can be assumed that the guesses follow a normal distribution.
  1. Estimate the mean and standard deviation of guesses.
  2. Five students are selected at random. Find the probability that at least three of them have a guess over $1.2$ kg.
24. In an A Level maths exam sat by $100$ students, $70$ score less than $80$%. Of these, $30$ score more than $40$%. Assuming the test scores are normally distributed, estimate the mean and standard deviation.