Given that $X\sim N(0,1)$, find:
- $\mathrm{P}(X < 0.5)$
- $\mathrm{P}(X > -0.1)$
- $0.691$
- $0.540$
Given that $X\sim N(1,0.5)$, find:
- $\mathrm{P}(X > 0 | X < 0.5)$
- $\mathrm{P}(X < 1.3 \cup X > 1.5)$
- $0.672$
- $0.904$
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.
Mean: $25$, standard deviation: $5$
Given that $X\sim N(5,10)$, find:
- $\mathrm{P}(6 < |X|)$
- $\mathrm{P}(X > 6 | 6 < |X|)$
- $0.376$
- $0.999$
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?
$147$
Given that $X\sim N(0,1)$, find $x$ such that:
- $\mathrm{P}(X < x) = 0.6$
- $\mathrm{P}(|X| < x) = 0.6$
- $0.253$
- $0.842$
A student's maths test score is normally distributed with mean $60$ and standard deviation $7$.
- What is the probability the student scores between $55$ and $65$ in a test?
- 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?
- What assumptions did you have to make in answering the previous question? Are these likely to be valid?
- $0.525$
- $0.0467$
- Results from each test are independent of the next and the probability of getting over $75$ is the same in each test. Unlikely because the student might get tired from doing three tests in one day.
The mean time a group of students can run a $100$m race is $16$ seconds, with a standard deviation of $1$ second.
- What is the probability that a randomly chosen student can run it in under $15$ seconds?
- What is the probability that a team of four students each run under $15$ seconds?
- 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?
- $0.159$
- $6.34\times 10^{-4}$
- More likely because when someone runs under 15 they allow someone else to run over 15 and still total less than a minute.
The heights of some students have a mean of $160$ cm, a median of $159$ cm, and a standard deviation of $5$ cm.
- Explain why a normal distribution could be a reasonable model for these data.
- What height should a student be to be in the tallest $10$% of students?
- The mean and the median are roughly the same.
- $166$
Given $X\sim N(\mu,1)$, find $\alpha$ such that $$\mathrm{P}(\mu - \alpha < X < \mu + \alpha) = 0.4$$
$0.524$
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 $x$.
$28.5$
Given $X\sim N(\mu,\sigma^2)$, find $\mu$ and $\sigma$ when:
- $\mathrm{P}(X > 3) = 0.6$ and $\mathrm{P}(X < 1) = 0.2$
- $\mathrm{P}(X > 20) = 0.3$ and $\mathrm{P}(X \geqslant 30) = 0.15$
- $\mu = 3.86$ and $\sigma = 3.40$
- $\mu = 9.76$ and $\sigma = 19.5$
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.
$\mu = 6.99$ and $\sigma = 1.92$
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.
$7.41$
The amount of pocket money some students earn in a year are shown below.
- What features of the diagram suggest a normal distribution is appropriate to model the amount of money students earn?
- Estimate the mean and standard deviation.
- A single mode, symmetrical, tapers off at the edges.
- Mean: $104$, Standard Deviation: $14.9$
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.
$N(60,42)$
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?
$0.512$
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.
- What is the probability the student is late for school on any given day?
- What is the probability the student is never late to school in one week?
- $0.0462$
- $0.790$
Given that $X \sim N(\mu,\sigma^2)$, find the interquartile range in terms of $\sigma$.
$1.35\sigma$
The following data show the amount of time, $t$ minutes, students spend on homework on an average evening:
| $t$ | Frequency |
|---|---|
| $0 < t \leqslant 10$ | $1$ |
| $10 < t \leqslant 20$ | $4$ |
| $20 < t \leqslant 30$ | $8$ |
| $30 < t \leqslant 40$ | $18$ |
| $40 < t \leqslant 50$ | $9$ |
| $50 < t \leqslant 60$ | $5$ |
- What features of this data make a normal distribution a suitable model?
- Assuming a normal distribution is appropriate, estimate the mean and standard deviation of times students spend on homework.
- Three students are selected at random. What is the probability they all spend more than $40$ minutes on homework in an average evening?
- Single mode, symmetrical, tapers off at edges
- $\mu = 35$, $\sigma = 11.738$
- $0.0376$
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:
- What is the standard deviation of this normal distribution?
- What is the probability that, out of a group of $10$ friends, fewer than $3$ spend more than $4$ hours on their phones?
- What assumptions did you have to make in the previous question, and are they likely to be valid?
- $1.0694$
- One person: $0.2272$ <br> $X \sim B(10, 0.2272)$ where $X$ is the number of friends who spend more than 4 hours on their phones <br> $\mathrm{P}(X < 3) = 0.5948$
- Assume the amount of time they spend on their phones is independent of each other, which is unlikely to be true if they are friends.
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.
- Estimate the mean and standard deviation of guesses.
- Five students are selected at random. Find the probability that at least three of them have a guess over $1.2$ kg.
- $\mu = 1040$, $\sigma = 283$
- $0.141$