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SL 4.9—Normal distribution - IB Questionbank | RevisionDojo

Practice SL 4.9—Normal distribution with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

The continuous random variable $X$ has probability density function $f$ given by

$$ f(x) = \begin{cases} 3ax, & 0 \leqslant x < 0.5 \\ a(2 - x), & 0.5 \leqslant x < 2 \\ 0, & \text{otherwise} \end{cases} $$

Show that $a = \frac{2}{3}$.

[3]

Find $P(X < 1)$.

[3]

Given that $P(s < X < 0.8) = 2 P(2s < X < 0.8)$, and that $0.25 < s < 0.4$, find the value of $s$.

[7]

Question 2

SLPaper 1

The masses of Fuji apples are normally distributed with a mean of 163 g and a standard deviation of 6.83 g. When Fuji apples are picked, they are classified as small, medium, large or extra large depending on their mass. Large apples have a mass of between 172 g and 183 g.

Determine the probability that a Fuji apple selected at random will be a large apple.

[2]

Approximately 68% of Fuji apples have a mass within the medium-sized category, which is between $k$ and 172 g. Find the value of $k$ .

[3]

Question 3

HLPaper 2

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ , such that $P(X < 30.31) = 0.1180$ and $P(X > 42.52) = 0.3060$ .

Find $\mu$ and $\sigma$ .

[6]

Find $P(|X - \mu| < 1.2\sigma)$ .

[2]

Question 4

HLPaper 3

Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.

7.7 \quad 7.5 \quad 8.4 \quad 8.8 \quad 7.3 \quad 9.0 \quad 7.8 \quad 7.6

Assume that these weights form a random sample from a $N(\mu, \sigma^2)$ distribution.

Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis $H_0: \mu = 7.5$.

Determine unbiased estimates for $\mu$ and $\sigma^2$.

[3]

Use a two-tailed test to determine the $p$-value for the above results.

[3]

Interpret your $p$-value at the 5% level of significance, justifying your conclusion.

[2]

Question 5

HLPaper 2

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean of 196 minutes and a standard deviation of 24 minutes.

It is found that 5% of the male runners complete the marathon in less than $T_1$ minutes.

The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean of 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.

Find the probability that a male runner selected at random will complete the marathon in less than 3 hours.

[2]

Calculate $T_1$.

[2]

Find the standard deviation of the times taken by female runners.

[4]

Question 6

HLPaper 1

A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.

Find $s_{n - 1}$ for this sample.

[2]

Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.

[2]

The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).

[1]

Question 7

HLPaper 1

The continuous random variable $X$ has a probability density function given by

f(x) = \begin{cases} k \sin \left( \dfrac{\pi x}{6} \right), & 0 \leqslant x \leqslant 6 \\ 0, & \text{otherwise} \end{cases}

Find the value of $k$ .

[4]

By considering the graph of $f$ , write down the mean of $X$ .

[1]

By considering the graph of $f$ , write down the median of $X$ .

[1]

By considering the graph of $f$ , write down the mode of $X$ .

[1]

Show that $P(0 \leqslant X \leqslant 2) = \dfrac{1}{4}$ .

[4]

Hence state the interquartile range of $X$ .

[3]

Calculate $P(X \leqslant 4 \mid X \geqslant 3)$ .

[2]

Question 8

HLPaper 2

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{ g}$ and standard deviation $8 \text{ g}$.

Find the probability that a randomly selected packet has a weight less than $100 \text{ g}$.

[2]

The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.

[2]

A packet is randomly selected. Given that the packet has a weight greater than $105 \text{ g}$, find the probability that it has a weight greater than $110 \text{ g}$.

[3]

From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.

[3]

Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{ g}$.

[4]

Question 9

HLPaper 1

Let $X$ be a random variable which follows a normal distribution with mean $\mu$ . Given that $P(X < \mu - 5) = 0.2$ , find

$P(X > \mu + 5)$ .

[2]

$P(X < \mu + 5 \mid X > \mu - 5)$ .

[5]

Question 10

HLPaper 3

In a large population of hens, the weight of a hen is normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. A random sample of 100 hens is taken from the population.

The mean weight for the sample is denoted by $\bar{X}$.

The sample values are summarized by $\sum x = 199.8$ and $\sum x^2 = 407.8$ where $x$ kg is the weight of a hen.

State the distribution of $\bar{X}$ giving its mean and variance.

[1]

Find an unbiased estimate for $\mu$.

[1]

Find an unbiased estimate for $\sigma^2$.

[2]

Find a 90% confidence interval for $\mu$.

[3]

It is found that $\sigma = 0.27$. It is decided to test, at the 1% level of significance, the null hypothesis $\mu = 1.95$ against the alternative hypothesis $\mu > 1.95$.

Find the $p$-value for the test.

[2]

Write down the conclusion reached.

[1]

Paper

Level

Question 1

HLPaper 2

The continuous random variable $X$ has probability density function $f$ given by

$$ f(x) = \begin{cases} 3ax, & 0 \leqslant x < 0.5 \\ a(2 - x), & 0.5 \leqslant x < 2 \\ 0, & \text{otherwise} \end{cases} $$

Show that $a = \frac{2}{3}$.

[3]

Find $P(X < 1)$.

[3]

Given that $P(s < X < 0.8) = 2 P(2s < X < 0.8)$, and that $0.25 < s < 0.4$, find the value of $s$.

[7]

Question 2

SLPaper 1

Determine the probability that a Fuji apple selected at random will be a large apple.

[2]

Approximately 68% of Fuji apples have a mass within the medium-sized category, which is between $k$ and 172 g. Find the value of $k$ .

[3]

Question 3

HLPaper 2

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ , such that $P(X < 30.31) = 0.1180$ and $P(X > 42.52) = 0.3060$ .

Find $\mu$ and $\sigma$ .

[6]

Find $P(|X - \mu| < 1.2\sigma)$ .

[2]

Question 4

HLPaper 3

Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.

7.7 \quad 7.5 \quad 8.4 \quad 8.8 \quad 7.3 \quad 9.0 \quad 7.8 \quad 7.6

Assume that these weights form a random sample from a $N(\mu, \sigma^2)$ distribution.

Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis $H_0: \mu = 7.5$.

Determine unbiased estimates for $\mu$ and $\sigma^2$.

[3]

Use a two-tailed test to determine the $p$-value for the above results.

[3]

Interpret your $p$-value at the 5% level of significance, justifying your conclusion.

[2]

Question 5

HLPaper 2

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean of 196 minutes and a standard deviation of 24 minutes.

It is found that 5% of the male runners complete the marathon in less than $T_1$ minutes.

Find the probability that a male runner selected at random will complete the marathon in less than 3 hours.

[2]

Calculate $T_1$.

[2]

Find the standard deviation of the times taken by female runners.

[4]

Question 6

HLPaper 1

A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.

Find $s_{n - 1}$ for this sample.

[2]

Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.

[2]

The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).

[1]

Question 7

HLPaper 1

The continuous random variable $X$ has a probability density function given by

f(x) = \begin{cases} k \sin \left( \dfrac{\pi x}{6} \right), & 0 \leqslant x \leqslant 6 \\ 0, & \text{otherwise} \end{cases}

Find the value of $k$ .

[4]

By considering the graph of $f$ , write down the mean of $X$ .

[1]

By considering the graph of $f$ , write down the median of $X$ .

[1]

By considering the graph of $f$ , write down the mode of $X$ .

[1]

Show that $P(0 \leqslant X \leqslant 2) = \dfrac{1}{4}$ .

[4]

Hence state the interquartile range of $X$ .

[3]

Calculate $P(X \leqslant 4 \mid X \geqslant 3)$ .

[2]

Question 8

HLPaper 2

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{ g}$ and standard deviation $8 \text{ g}$.

Find the probability that a randomly selected packet has a weight less than $100 \text{ g}$.

[2]

The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.

[2]

A packet is randomly selected. Given that the packet has a weight greater than $105 \text{ g}$, find the probability that it has a weight greater than $110 \text{ g}$.

[3]

From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.

[3]

Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{ g}$.

[4]

Question 9

HLPaper 1

Let $X$ be a random variable which follows a normal distribution with mean $\mu$ . Given that $P(X < \mu - 5) = 0.2$ , find

$P(X > \mu + 5)$ .

[2]

$P(X < \mu + 5 \mid X > \mu - 5)$ .

[5]

Question 10

HLPaper 3

In a large population of hens, the weight of a hen is normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. A random sample of 100 hens is taken from the population.

The mean weight for the sample is denoted by $\bar{X}$.

The sample values are summarized by $\sum x = 199.8$ and $\sum x^2 = 407.8$ where $x$ kg is the weight of a hen.

State the distribution of $\bar{X}$ giving its mean and variance.

[1]

Find an unbiased estimate for $\mu$.

[1]

Find an unbiased estimate for $\sigma^2$.

[2]

Find a 90% confidence interval for $\mu$.

[3]

It is found that $\sigma = 0.27$. It is decided to test, at the 1% level of significance, the null hypothesis $\mu = 1.95$ against the alternative hypothesis $\mu > 1.95$.

Find the $p$-value for the test.

[2]

Write down the conclusion reached.

[1]

Paper

Level

Question 1

HLPaper 2

The continuous random variable $X$ has probability density function $f$ given by

$$ f(x) = \begin{cases} 3ax, & 0 \leqslant x < 0.5 \\ a(2 - x), & 0.5 \leqslant x < 2 \\ 0, & \text{otherwise} \end{cases} $$

Show that $a = \frac{2}{3}$.

[3]

Find $P(X < 1)$.

[3]

Given that $P(s < X < 0.8) = 2 P(2s < X < 0.8)$, and that $0.25 < s < 0.4$, find the value of $s$.

[7]

Question 2

SLPaper 1

Determine the probability that a Fuji apple selected at random will be a large apple.

[2]

Approximately 68% of Fuji apples have a mass within the medium-sized category, which is between $k$ and 172 g. Find the value of $k$ .

[3]

Question 3

HLPaper 2

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ , such that $P(X < 30.31) = 0.1180$ and $P(X > 42.52) = 0.3060$ .

Find $\mu$ and $\sigma$ .

[6]

Find $P(|X - \mu| < 1.2\sigma)$ .

[2]

Question 4

HLPaper 3

Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.

7.7 \quad 7.5 \quad 8.4 \quad 8.8 \quad 7.3 \quad 9.0 \quad 7.8 \quad 7.6

Assume that these weights form a random sample from a $N(\mu, \sigma^2)$ distribution.

Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis $H_0: \mu = 7.5$.

Determine unbiased estimates for $\mu$ and $\sigma^2$.

[3]

Use a two-tailed test to determine the $p$-value for the above results.

[3]

Interpret your $p$-value at the 5% level of significance, justifying your conclusion.

[2]

Question 5

HLPaper 2

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean of 196 minutes and a standard deviation of 24 minutes.

It is found that 5% of the male runners complete the marathon in less than $T_1$ minutes.

Find the probability that a male runner selected at random will complete the marathon in less than 3 hours.

[2]

Calculate $T_1$.

[2]

Find the standard deviation of the times taken by female runners.

[4]

Question 6

HLPaper 1

A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.

Find $s_{n - 1}$ for this sample.

[2]

Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.

[2]

The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).

[1]

Question 7

HLPaper 1

The continuous random variable $X$ has a probability density function given by

f(x) = \begin{cases} k \sin \left( \dfrac{\pi x}{6} \right), & 0 \leqslant x \leqslant 6 \\ 0, & \text{otherwise} \end{cases}

Find the value of $k$ .

[4]

By considering the graph of $f$ , write down the mean of $X$ .

[1]

By considering the graph of $f$ , write down the median of $X$ .

[1]

By considering the graph of $f$ , write down the mode of $X$ .

[1]

Show that $P(0 \leqslant X \leqslant 2) = \dfrac{1}{4}$ .

[4]

Hence state the interquartile range of $X$ .

[3]

Calculate $P(X \leqslant 4 \mid X \geqslant 3)$ .

[2]

Question 8

HLPaper 2

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{ g}$ and standard deviation $8 \text{ g}$.

Find the probability that a randomly selected packet has a weight less than $100 \text{ g}$.

[2]

The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.

[2]

A packet is randomly selected. Given that the packet has a weight greater than $105 \text{ g}$, find the probability that it has a weight greater than $110 \text{ g}$.

[3]

From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.

[3]

Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{ g}$.

[4]

Question 9

HLPaper 1

Let $X$ be a random variable which follows a normal distribution with mean $\mu$ . Given that $P(X < \mu - 5) = 0.2$ , find

$P(X > \mu + 5)$ .

[2]

$P(X < \mu + 5 \mid X > \mu - 5)$ .

[5]

Question 10

HLPaper 3

In a large population of hens, the weight of a hen is normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. A random sample of 100 hens is taken from the population.

The mean weight for the sample is denoted by $\bar{X}$.

The sample values are summarized by $\sum x = 199.8$ and $\sum x^2 = 407.8$ where $x$ kg is the weight of a hen.

State the distribution of $\bar{X}$ giving its mean and variance.

[1]

Find an unbiased estimate for $\mu$.

[1]

Find an unbiased estimate for $\sigma^2$.

[2]

Find a 90% confidence interval for $\mu$.

[3]

It is found that $\sigma = 0.27$. It is decided to test, at the 1% level of significance, the null hypothesis $\mu = 1.95$ against the alternative hypothesis $\mu > 1.95$.

Find the $p$-value for the test.

[2]

Write down the conclusion reached.

[1]

SL 4.9—Normal distribution Questionbank