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Date	May Example question	Marks available	8	Reference code	EXM.3.AHL.TZ0.8
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	State and Determine	Question number	8	Adapted from	N/A

Question

This question explores methods to analyse the scores in an exam.

A random sample of 149 scores for a university exam are given in the table.

The university wants to know if the scores follow a normal distribution, with the mean and variance found in part (a).

The expected frequencies are given in the table.

The university assigns a pass grade to students whose scores are in the top 80%.

The university also wants to know if the exam is gender neutral. They obtain random samples of scores for male and female students. The mean, sample variance and sample size are shown in the table.

The university awards a distinction to students who achieve high scores in the exam. Typically, 15% of students achieve a distinction. A new exam is trialed with a random selection of students on the course. 5 out of 20 students achieve a distinction.

A different exam is trialed with 16 students. Let $p$ be the percentage of students achieving a distinction. It is desired to test the hypotheses

${H_0}\,{\text{:}}\,p = 0.15$ against ${H_1}\,{\text{:}}\,p > 0.15$

It is decided to reject the null hypothesis if the number of students achieving a distinction is greater than 3.

Find unbiased estimates for the population mean.

[1]

a.i.

Find unbiased estimates for the population Variance.

[2]

a.ii.

Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.

[3]

Perform a suitable test, at the 5% significance level, to determine if the scores follow a normal distribution, with the mean and variance found in part (a). You should clearly state your hypotheses, the degrees of freedom, the p-value and your conclusion.

[8]

Use the normal distribution model to find the score required to pass.

[2]

Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.

[6]

Perform a suitable test, at the 5% significance level, to determine if it is easier to achieve a distinction on the new exam. You should clearly state your hypotheses, the critical region and your conclusion.

[6]

Find the probability of making a Type I error.

[3]

g.i.

Given that $p = 0.2$ find the probability of making a Type II error.

[3]

g.ii.

Markscheme

52.8 A1

[1 mark]

a.i.

$s_{n - 1}^2 = {23.7^2} = 562$ M1A1

[2 marks]

a.ii.

$P\left( {20 < x \leqslant 40} \right) = 0.211$ M1A1

$0.211 \times 149$ M1

= 31.5 AG

[3 marks]

use of a ${\chi ^2}$ goodness of fit test M1

${H_0}\,{\text{:}}\,x \sim N\left( {52.8{\text{,}}\,\,562} \right)$ and A1A1

$\upsilon = 5 - 1 - 2 = 2$ A1

p-value = 0.569 A2

Since 0.569 > 0.05 R1

Insufficient evidence to reject ${H_0}$ . The scores follow a normal distribution. A1

[8 marks]

${\Phi ^{ - 1}}\left( {0.2} \right) = 32.8$ M1A1

[2 marks]

use of a t-test M1

${H_0}\,{\text{:}}\,{\mu _m} = {\mu _f}$ and ${H_1}\,{\text{:}}\,{\mu _m} \ne {\mu _f}$ A1

p-value = 0.180 A2

Since 0.180 > 0.05 R1

Insufficient evidence to reject ${H_0}$ . There is no difference between males and females. A1

[6 marks]

use of test for proportion using Binomial distribution M1

${H_0}\,{\text{:}}\,p = 0.15$ and ${H_1}\,{\text{:}}\,p > 0.15$ A1

$P\left( {X \geqslant 6} \right) = 0.0673$ and $P\left( {X \geqslant 7} \right) = 0.0219$ M1

So the critical region is ${X \geqslant 7}$ A1

Since 5 < 7 R1

Insufficient evidence to reject ${H_0}$ . It is not easier to achieve a distinction on the new exam. A1

[6 marks]

using ${H_0}{\text{,}}\,\,X \sim B\left( {16{\text{,}}\,\,0.15} \right)$ M1

$P\left( {X > 3} \right) = 0.210$ M1A1

[3 marks]

g.i.

using ${H_1}{\text{,}}\,\,X \sim B\left( {16{\text{,}}\,\,0.2} \right)$ M1

$P\left( {X \leqslant 3} \right) = 0.598$ M1A1

[3 marks]

g.ii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

g.i.

[N/A]

g.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations

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17N.2.SL.TZ0.S_7:
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EXM.2.AHL.TZ0.26b:
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EXM.2.AHL.TZ0.27a.ii:
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19N.1.SL.TZ0.T_12a:
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19M.1.SL.TZ1.S_9d:
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Find the mean and standard deviation of the sample data in the table above. Show how you arrived at your answers.
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18M.2.SL.TZ1.S_9b.ii:
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17M.1.SL.TZ2.S_3a:
Write down ${\text{P}}(X > 107)$ .
19N.1.SL.TZ0.T_12c:
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Find the value of $k$ .
19M.2.AHL.TZ2.H_2b:
The shop is open for 24 days every month.

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18M.2.SL.TZ1.S_9d:
Find the probability that the grocer buys more than half the oranges in a box selected at random.
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Calculate the new value of $\sigma$ .
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Find the standardized value for 289 g.
18N.2.AHL.TZ0.H_3b:
Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.
17N.1.SL.TZ0.T_13a:
Find the probability that this applicant took at least 40 minutes to complete the test.
16N.2.SL.TZ0.T_4b:
(i) Find the probability that a box of cereal, chosen at random, is sold.

(ii) Calculate the manufacturer’s expected daily income from these sales.
18N.3.AHL.TZ0.Hsp_3d:
Use an appropriate regression line to estimate the weight of a fish with length 360 mm.
18M.2.AHL.TZ2.H_3a:
Sketch the probability density function for X, and shade the region representing P(μ − 2σ < X < μ + σ).
18M.2.AHL.TZ2.H_3c:
Find the value of k for which P(μ − kσ < X < μ + kσ) = 0.5.
EXM.3.AHL.TZ0.8b:
Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.
21M.2.SL.TZ2.4c:
Determine the expected number of cats in this group that have a weight of less than $5.3 kg$ .
18N.2.SL.TZ0.S_9d:
Ten of the participants with reaction times greater than 0.65 are selected at random. Find the probability that at least two of them are in Group X.
17M.1.SL.TZ2.T_11b:
Find the value of $a$ .
16N.2.SL.TZ0.T_4a:
Draw a diagram that shows this information.
18N.1.SL.TZ0.T_14a:
Find the value of k.
EXM.3.AHL.TZ0.8g.i:
Find the probability of making a Type I error.
17M.2.AHL.TZ1.H_9a:
Find the probability that a runner selected at random will complete the marathon in less than 3 hours.
17M.2.AHL.TZ2.H_3b:
Calculate the new value of $\mu$ giving your answer correct to two decimal places.
19M.1.SL.TZ1.S_9a:
Find P(−1.6 < $z$ < 2.4). Write your answer in terms of $a$ and $b$ .
17M.2.SL.TZ1.S_9b:
Find the value of $\mu$ .
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Find the probability that a melon selected at random will have a mass greater than $3.0 kg$ .
18N.3.AHL.TZ0.Hsp_3a.i:
Find the probability that a fish from this lake will have a weight of more than 560 grams.
18N.3.AHL.TZ0.Hsp_3b.iii:
State the conclusion of the test, justifying your answer.
16N.2.AHL.TZ0.H_8a:
Find $\mu$ and $\sigma$ .
17N.2.AHL.TZ0.H_4:
It is given that one in five cups of coffee contain more than 120 mg of caffeine.
It is also known that three in five cups contain more than 110 mg of caffeine.

Assume that the caffeine content of coffee is modelled by a normal distribution.
Find the mean and standard deviation of the caffeine content of coffee.
EXM.1.AHL.TZ0.59a:
Calculate unbiased estimates of the population mean and the population variance.
20N.2.SL.TZ0.T_6a:
Sketch a diagram to represent this information.
20N.2.SL.TZ0.T_6b:
Show that $m = 15.7$ .
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Find the probability that Emlyn plays between $13 minutes$ and $18 minutes$ in a game.
20N.2.SL.TZ0.T_6c.ii:
Find the probability that Emlyn plays more than $20 minutes$ in a game.
20N.2.SL.TZ0.T_6d:
Find the value of $x$ .
20N.2.SL.TZ0.T_6f:
Find the expected number of successful shots Emlyn will make on Monday, based on the results from Saturday and Sunday.
20N.2.SL.TZ0.T_6g:
Emlyn claims the results from Saturday and Sunday show that his expected number of successful shots will be more than Johan’s.

Determine if Emlyn’s claim is correct. Justify your reasoning.
16N.2.SL.TZ0.T_4c:
Calculate the manufacturer’s expected daily recycling cost.
17M.2.AHL.TZ1.H_9c:
Find the standard deviation of the times taken by female runners.
18N.3.AHL.TZ0.Hsp_3c.ii:
Find the product-moment correlation coefficient $r$ .
19M.1.AHL.TZ1.H_6a:
${\text{P}}\left( {X > \mu + 5} \right)$ .
18M.2.SL.TZ2.S_10a.ii:
Show that μ = 106.
18M.2.SL.TZ2.T_3d.i:
For this test state the null hypothesis.
18N.3.AHL.TZ0.Hsp_3a.ii:
The maximum weight a hand net can hold is 6 kg. Find the probability that a catch of 11 fish can be carried in the hand net.
19M.1.SL.TZ1.S_9b:
Given that $z$ > −1.6, find the probability that z < 2.4 . Write your answer in terms of $a$ and $b$ .
17N.1.SL.TZ0.T_13b:
Find the value of $k$ .
18N.3.AHL.TZ0.Hsp_3b.ii:
Find the p-value for the test.
17M.2.SL.TZ1.T_5f.ii:
Calculate the expected number of students that spent at least 90 minutes preparing for the test.
19M.2.SL.TZ2.S_9d:
All flights have two pilots. Find the percentage of flights where both pilots flew more than 30 hours last week.
17M.2.AHL.TZ2.H_3a:
Given that $\mu = 253$ and $\sigma = 1.5$ find the probability that a randomly chosen packet of biscuits is underweight.
17M.1.SL.TZ2.S_3b:
Find ${\text{P}}(100 < X < 107)$ .
19M.2.SL.TZ2.S_9b.i:
Find ${\text{P}}\left( {A \cap D'} \right)$ .
18M.2.SL.TZ2.S_10b:
Find P(M < 95) .
21M.2.AHL.TZ2.2b:
Find the proportion of male Persian cats weighing between $5.5 kg$ and $6.5 kg$ .
21M.2.AHL.TZ2.2c:
Determine the expected number of cats in this group that have a weight of less than $5.3 kg$ .
18M.2.SL.TZ1.S_9e:
The grocer selects two boxes at random.

Find the probability that the grocer buys more than half the oranges in each box.
16N.2.SL.TZ0.S_5b:
A newborn baby has a low birth weight.

Find the probability that the baby weighs at least 2.15 kg.
17M.2.SL.TZ1.T_5a.i:
Calculate the mean test grade of the students;
EXM.2.AHL.TZ0.26a:
The scientists wish to investigate the claim that Group B gain weight faster than Group A. Test this claim at the 5% level of significance, noting which hypothesis test you are using. You may assume that the weight gain for each group is normally distributed, with the same variance, and independent from each other.
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Find the probability that a bag of apples selected at random contains at most one small apple.
17M.1.SL.TZ2.S_3c:
Find ${\text{P}}(93 < X < 107)$ .
21M.2.SL.TZ1.4a.i:
in less than $6.5 m$ .
21M.2.SL.TZ1.4a.ii:
in more than $7 m$ .
18M.2.SL.TZ2.T_3d.ii:
For this test find the p-value.
21M.2.SL.TZ1.4b.i:
$6.5 m$ and $6.75 m$ .
18M.2.AHL.TZ1.H_4b:
Eight wolves are independently selected at random and their ages recorded.

Find the probability that more than six of these wolves are at least 5 years old.
19M.2.SL.TZ2.S_9c:
Find the value of $\sigma$ .
18M.2.SL.TZ2.S_10a.i:
Write down the value of k.
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Estimate the number of applicants who completed the test in less than 25 minutes.
EXM.2.AHL.TZ0.27a.i:
a goodness of fit test (a complete explanation required);
16N.2.SL.TZ0.S_5a:
Given that 5.3% of newborn babies have a low birth weight, find $w$ .
EXM.2.AHL.TZ0.27c:
Most of the expected frequencies have been calculated in the third column. (Frequencies have been rounded to the nearest integer, and frequencies in the first and last classes have been extended to include the rest of the data beyond 15 and 225. Find the values of $a$ , $b$ and $c$ and show how you arrived at your answers.
17M.2.SL.TZ1.S_9a:
Find ${\text{P}}(X < 9)$ .
21M.2.SL.TZ2.4b:
Find the proportion of male Persian cats weighing between $5.5 kg$ and $6.5 kg$ .
21M.2.AHL.TZ2.4a:
Write down a transition matrix $T$ indicating the annual population movement between clinics.
18N.1.SL.TZ0.T_14c:
Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.
19M.1.AHL.TZ1.H_6b:
${\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)$ .
18M.2.SL.TZ2.T_3e:
State a conclusion for this test. Justify your answer.
EXN.1.SL.TZ0.3a:
Find the probability that an apple from the tree has a weight greater than $90$ grams.
18N.2.AHL.TZ0.H_3a:
Find the value of $\mu$ and the value of $\sigma$ .
EXM.2.AHL.TZ0.27e:
State your hypotheses, critical number, decision rule and conclusion (using a 5% level of significance).
EXM.3.AHL.TZ0.8e:
Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.
EXM.3.AHL.TZ0.8g.ii:
Given that $p = 0.2$ find the probability of making a Type II error.
17M.2.SL.TZ1.T_5f.i:
Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.
21M.2.AHL.TZ2.4b:
Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.
21M.1.AHL.TZ1.14c:
Find the probability that a bag selected at random weighs more than $1 kg$ .
21M.2.SL.TZ1.4b.ii:
$6.75 m$ and $7 m$ .
21M.1.AHL.TZ1.14b:
Find the standard deviation of the weights of these bags of apples.
21M.2.SL.TZ2.4d:
It is found that $12$ of the cats weigh more than $x kg$ . Estimate the value of $x$ .
21M.2.AHL.TZ2.4c:
Find a matrix $P$ , with integer elements, such that $T = P D P^{- 1}$ , where $D$ is a diagonal matrix.
21M.2.AHL.TZ2.4d:
Hence, show that the long-term transition matrix $T^{\infty}$ is given by $T^{\infty} = (\begin{matrix} \frac{10}{17} & \frac{10}{17} \\ \frac{7}{17} & \frac{7}{17} \end{matrix})$ .
21M.2.AHL.TZ2.4e:
Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.
21N.1.AHL.TZ0.14c:
The sacks of potatoes are transported in crates. There are $10$ sacks in each crate and the weights of the sacks of potatoes are independent of each other.

Find the probability that the total weight of the sacks of potatoes in a crate exceeds $500 kg$ .
21N.2.SL.TZ0.5d.i:
Find the probability that Arianne scores at least $5$ points in the competition.
EXN.2.AHL.TZ0.3b:
Find the value of $a$ and the value of $b$ , giving your answers correct to one decimal place.
21N.2.AHL.TZ0.3d.iv:
Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.
21N.2.SL.TZ0.5a.i:
a dart lands less than $13 cm$ from $O$ .
21N.2.SL.TZ0.5a.ii:
a dart lands more than $15 cm$ from $O$ .
21N.2.SL.TZ0.5c:
Find the probability that Arianne does not score a point on a turn of three darts.
21N.2.SL.TZ0.5d.ii:
Find the probability that Arianne scores at least $5$ points and less than $8$ points.
21N.2.SL.TZ0.5d.iii:
Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.
21N.2.AHL.TZ0.3a.i:
a dart lands less than $13 cm$ from $O$ .
21N.2.AHL.TZ0.3a.ii:
a dart lands more than $15 cm$ from $O$ .
21N.2.AHL.TZ0.3c:
Find the probability that Arianne does not score a point on a turn of three darts.
21N.2.AHL.TZ0.3d.i:
Find Arianne’s expected score in the competition.
21N.2.AHL.TZ0.3d.ii:
Find the probability that Arianne scores at least $5$ points in the competition.
21N.2.AHL.TZ0.3d.iii:
Find the probability that Arianne scores at least $5$ points and less than $8$ points.

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Topic 4—Statistics and probability

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