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Date	May 2022	Marks available	6	Reference code	22M.3.AHL.TZ2.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 2
Command term	Find	Question number	1	Adapted from	N/A

Question

This question uses statistical tests to investigate whether advertising leads to increased profits for a grocery store.

Aimmika is the manager of a grocery store in Nong Khai. She is carrying out a statistical analysis on the number of bags of rice that are sold in the store each day. She collects the following sample data by recording how many bags of rice the store sells each day over a period of $90$ days.

She believes that her data follows a Poisson distribution.

Aimmika knows from her historic sales records that the store sells an average of $4.2$ bags of rice each day. The following table shows the expected frequency of bags of rice sold each day during the $90$ day period, assuming a Poisson distribution with mean $4.2$ .

Aimmika decides to carry out a $χ^{2}$ goodness of fit test at the $5 %$ significance level to see whether the data follows a Poisson distribution with mean $4.2$ .

Aimmika claims that advertising in a local newspaper for $300$ Thai Baht $(THB)$ per day will increase the number of bags of rice sold. However, Nichakarn, the owner of the store, claims that the advertising will not increase the store’s overall profit.

Nichakarn agrees to advertise in the newspaper for the next $60$ days. During that time, Aimmika records that the store sells $282$ bags of rice with a profit of $495 THB$ on each bag sold.

Aimmika wants to carry out an appropriate hypothesis test to determine whether the number of bags of rice sold during the $60$ days increased when compared with the historic sales records.

Find the mean and variance for the sample data given in the table.

[2]

a.i.

Hence state why Aimmika believes her data follows a Poisson distribution.

[1]

a.ii.

State one assumption that Aimmika needs to make about the sales of bags of rice to support her belief that it follows a Poisson distribution.

[1]

Find the value of $a$ , of $b$ , and of $c$ . Give your answers to $3$ decimal places.

[5]

Write down the number of degrees of freedom for her test.

[1]

d.i.

Perform the $χ^{2}$ goodness of fit test and state, with reason, a conclusion.

[7]

d.ii.

By finding a critical value, perform this test at a $5 %$ significance level.

[6]

e.i.

Hence state the probability of a Type I error for this test.

[1]

e.ii.

By considering the claims of both Aimmika and Nichakarn, explain whether the advertising was beneficial to the store.

[3]

Markscheme

mean $= 4.23 (4.23333 \dots)$ A1

variance $= 4.27 (4.26777 \dots)$ A1

[2 marks]

a.i.

mean is close to the variance A1

[1 mark]

a.ii.

One of the following:

the number of bags sold each day is independent of any other day

the sale of one bag is independent of any other bag sold

the sales of bags of rice (each day) occur at a constant mean rate A1

Note: Award A1 for a correct answer in context. Any statement referring to independence must refer to either the independence of each bag sold or the independence of the number of bags sold each day. If the third option is seen, the statement must refer to a “constant mean” or “constant average”. Do not accept “the number of bags sold each day is constant”.

[1 mark]

attempt to find Poisson probabilities and multiply by $90$ (M1)

$a = 7.018$ A1

$b = 17.498$ A1

EITHER

$90 \times P (X \geq 8) = 90 \times (1 - P (X \leq 7))$ (M1)

$c = 5.755$ A1

$90 - 7.018 - 11.903 - 16.665 - 17.498 - 14.698 - 10.289 - 6.173$ (M1)

$c = 5.756$ A1

Note: Do not penalize the omission of clear $a$ , $b$ and $c$ labelling as this will be penalized later if correct values are interchanged.

[5 marks]

$7$ A1

[1 mark]

d.i.

$H_{0} :$ The number of bags of rice sold each day follows a Poisson distribution with mean $4.2$ . A1

$H_{1} :$ The number of bags of rice sold each day does not follow a Poisson distribution with mean $4.2$ . A1

Note: Award A1A1 for both hypotheses correctly stated and in correct order. Award A1A0 if reference to the data and/or “mean $4.2$ ” is not included in the hypotheses, but otherwise correct.

evidence of attempting to group data to obtain the observed frequencies for $\leq 1$ and $\geq 8$ (M1)

$p$ -value $= 0.728 (0.728100 \dots)$ A2

$0.728 (0.728100 \dots) > 0.05$ R1

the result is not significant so there is no reason to reject $H_{0}$ (the number of bags sold each day follows a Poisson distribution) A1

Note: Do not award R0A1. The conclusion MUST follow through from their hypotheses. If no hypotheses are stated, the final A1 can still be awarded for a correct conclusion as long as it is in context (e.g. therefore the data follows a Poisson distribution).

[7 marks]

d.ii.

METHOD 1

evidence of multiplying $4.2 \times 60$ (seen anywhere) M1

$H_{0} : μ = 252$

$H_{1} : μ > 252$ A1

Note: Accept $H_{0} : μ = 4.2$ and $H_{1} : μ > 4.2$ for the A1.

evidence of finding probabilities around critical region (M1)

Note: Award (M1) for any of these values seen:

$P (X \geq 277) = 0.0630518 \dots$ OR $P (X \leq 276) = 0.936948 \dots$

$P (X \geq 278) = 0.0558415 \dots$ OR $P (X \leq 277) = 0.944158 \dots$

$P (X \geq 279) = 0.0493055 \dots$ OR $P (X \leq 278) = 0.950694 \dots$

critical value $= 279$ A1

$282 \geq 279$ , R1

the null hypothesis is rejected A1

(the advertising increased the number of bags sold during the $60$ days)

Note: Do not award R0A1. Accept statements referring to the advertising being effective for A1 as long as the R mark is satisfied. For the R1A1, follow through within the part from their critical value.

METHOD 2

evidence of dividing $282$ by $60$ (or $4.7$ seen anywhere) M1

$H_{0} : μ = 4.2$

$H_{1} : μ > 4.2$ A1

attempt to find critical value using central limit theorem (M1)

(e.g. sample standard deviation $= \sqrt{\frac{4.2}{60}}, X ~ N (4.2, \sqrt{\frac{4.2}{60}})$ , etc.)

Note: Award (M1) for a $p$ -value of $0.0293907 \dots$ seen.

critical value $= 4.63518 \dots$ A1

$4.7 > 4.63518 \dots$ R1

the null hypothesis is rejected A1

(the advertising increased the number of bags sold during the $60$ days)

[6 marks]

e.i.

$(P (X \geq 279 μ = 252) =) 0.0493 (0.0493055 \dots)$ A1

Note: If a candidate uses METHOD 2 in part (e)(i), allow an FT answer of $0.05$ for this part but only if the candidate has attempted to find a $p$ -value.

[1 mark]

e.ii.

attempt to compare profit difference with cost of advertising (M1)

Note: Award (M1) for evidence of candidate mathematically comparing a profit difference with the cost of the advertising.

EITHER

(comparing profit from $30$ extra bags of rice with cost of advertising)
$14850 < 18000$ A1

OR

(comparing total profit with and without advertising)
$121590 < 124740$ A1

(comparing increase of average daily profit with daily advertising cost)
$247.50 < 300$ A1

THEN

EITHER

Even though the number of bags of rice increased, the advertising is not worth it as the overall profit did not increase. R1

The advertising is worth it even though the cost is less than the increased profit, since the number of customers increased (possibly buying other products and/or returning in the future after advertising stops) R1

Note: Follow through within the part for correct reasoning consistent with their comparison.

[3 marks]

Examiners report

Candidates generally did well in finding the mean, although some wasted time by calculating it by hand rather than by using their GDC. Many candidates were able to find a correct variance. However, there were also many who gave the standard deviation as their variance or simply made the variance the same as their mean without performing a calculation, possibly looking ahead to part (a)(ii). Many candidates successfully used the clue given by the command term “hence” and correctly answered (a)(ii).

a.i.

[N/A]

a.ii.

It was clear candidates understood that independence was the key term needed in the response. However, a number of candidates struggled either by not being precise enough in their responses (e.g. simply stating “they are independent”) or by incorrectly stating that the bags of rice sold must be independent of the number of days. “Communication” is an assessment objective for the course, and candidates should aim for clarity in their responses thereby ensuring the examiner can be confident in awarding credit.

This question was generally done well by the candidates. The two most common mistakes both stemmed from candidates not paying attention to the instructions given. Candidates either did not give their answer correct to 3 decimal places or they incorrectly attempted to use the normal distribution to find the expected frequencies. Another frequent mistake involved candidates multiplying their probabilities by 100 rather than by 90.

While most candidates were able to gain the mark in part (i), many candidates arrived at a correct answer but by using the incorrect 𝜒² test for independence method of determining the degrees of freedom. In part (ii), several common mistakes led to very few candidates receiving the full seven marks for this question part. Candidates struggled to correctly write the hypotheses as they either wrote them in reverse order or they did not correctly reference the data and/or a Poisson distribution with mean 4.2. Unfortunately, some did not state the hypotheses at all. Another common mistake involved candidates incorrectly combining columns to create a column for days when at least 7 bags of rice were sold, possibly from incorrectly thinking that an observed value less than 5 is not allowed when carrying out goodness of fit test. Many candidates also missed out on possible follow through marks for their p-value by not fully writing out the observed and expected values they were inputting into their GDC. Although communication was not being assessed here, it highlights how it is easier to credit a correct method (even if leading to an incorrect answer) if there is appropriate working and/or a running commentary present.

d.i.

[N/A]

d.ii.

In contrast to part 1(d) where a method was given, many candidates struggled to know how to find a critical value in this question part. Although it was possible for candidates to find a critical value by either using Poisson probabilities or probabilities from a normal approximation, few knew how to begin. As a result, very few candidates scored full marks here.

e.i.

[N/A]

e.ii.

Very few candidates managed to get full marks in this question part. Again, without being guided into a particular method, candidates struggled to understand how to begin the problem. Many candidates did not realize that some calculations were necessary to give a proper conclusion. A subset of these candidates thought it was a continuation of part (e) and made some statement related to their conclusion from the hypothesis test. For the candidates that did try to make some calculations, many simply calculated the profit from selling 282 bags of rice and compared that to the cost of the advertising. These candidates did not realize they needed to compare the difference in the expected profit without advertising and the actual profit with advertising.

Syllabus sections

Topic 4—Statistics and probability » SL 4.11—Expected, observed, hypotheses, chi squared, gof, t-test

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EXM.1.AHL.TZ0.59a:
Calculate unbiased estimates of the population mean and the population variance.
19N.2.SL.TZ0.T_1f.i:
Calculate the probability that the customer is an adult.
17N.2.SL.TZ0.T_1d:
Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
18N.1.SL.TZ0.T_4c:
Find the p-value for this test.
20N.1.SL.TZ0.T_10b:
Find the expected number of short trips when it rained.
20N.1.SL.TZ0.T_10c:
The $p$ -value for this test is $0.0206$ .

State the conclusion to Isaac’s test. Justify your reasoning.
EXN.1.SL.TZ0.3b.ii:
The test is performed at the $5 %$ significance level.

State the conclusion of the test, giving a reason for your answer.
EXN.1.SL.TZ0.3b.i:
Find the $p$ -value for the owner’s test.
EXM.1.SL.TZ0.10b:
Explain what is meant by “level of significance” in part (a).
EXM.2.AHL.TZ0.28b.i:
Find the significance level of this procedure.
16N.1.SL.TZ0.T_6a:
Find the expected frequency of the patients who became infected whilst in Nightingale ward.
17M.2.SL.TZ2.T_1b:
The teacher groups the responses by topic, and chooses two responses to the Logic question. Find the probability that both are not satisfactory.
EXN.2.SL.TZ0.4c.iii:
State the degrees of freedom for the test.
18M.2.SL.TZ2.T_3d.i:
For this test state the null hypothesis.
16N.1.SL.TZ0.T_6c:
State, giving a reason, whether the null hypothesis should be rejected.
18N.1.SL.TZ0.T_4d:
Abhinav rejects H₀.

State a reason why Abhinav is incorrect in doing so.
EXM.1.AHL.TZ0.58a:
Find the exact value of the mean of this distribution.
EXM.2.AHL.TZ0.28b.ii:
Some time later, the actual value of $\mu$ is 503. Find the probability of a Type II error.
19M.2.SL.TZ1.T_3e.i:
State the null hypothesis.
EXM.2.AHL.TZ0.24a:
Show that this data leads to an estimated value of $p = 0.4$ .
20N.1.SL.TZ0.T_10a:
Write down $H_{0}$ , the null hypothesis for this test.
18M.2.SL.TZ1.T_2e:
State, with a reason, whether you would reject the null hypothesis.
21M.1.SL.TZ1.6b:
Calculate the $p$ -value for this test.
19M.2.SL.TZ1.T_3c.i:
For these data, write down the median number of pets.
19M.2.SL.TZ1.T_3c.ii:
For these data, write down the lower quartile.
SPM.3.AHL.TZ0.1d:
Use a paired $t$ -test to determine whether there is significant evidence that the students in school A have improved their IB points since the start of the course.
17M.1.SL.TZ1.T_6c:
Use your graphic display calculator to find the ${\chi ^2}$ statistic for this test.
17M.2.SL.TZ2.T_1a.iii:
If the teacher chooses a response at random, find the probability that it is a satisfactory response, given that it is a response to the Calculus question.
17M.2.SL.TZ2.T_1g:
State the conclusion of this ${\chi ^2}$ test. Give a reason for your answer.
17M.2.SL.TZ2.T_1f:
Use your graphic display calculator to find the ${\chi ^2}$ statistic for this data.
21M.1.SL.TZ1.6a:
State, in words, the null hypothesis.
21M.1.SL.TZ1.6c:
State whether the result of the test supports Arriane’s claim. Justify your reasoning.
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.
18M.2.SL.TZ1.T_2b:
Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
EXM.2.AHL.TZ0.28a.ii:
Calculate the mean of these data and hence estimate the value of $p$ .
21M.2.SL.TZ1.4c:
State the null and alternative hypotheses.
21M.2.SL.TZ1.4d:
Find the $p$ -value for the test.
18M.2.SL.TZ2.T_3d.ii:
For this test find the p-value.
18N.1.SL.TZ0.T_4b:
Calculate the expected number of male engineers.
EXM.2.AHL.TZ0.27a.i:
a goodness of fit test (a complete explanation required);
21M.1.SL.TZ2.8c:
Write down the conclusion to the test. Give a reason for your answer.
18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
18M.2.SL.TZ1.T_2h:
Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.
21M.1.SL.TZ2.11b.i:
Write down the degrees of freedom for this test.
21M.1.SL.TZ2.11a:
Find an estimate for how many copies the vendor expects to sell each day.
21M.1.SL.TZ2.11b.ii:
Write down the conclusion to the test. Give a reason for your answer.
21M.1.SL.TZ2.8b:
Find the $p$ -value for this test.
EXM.1.AHL.TZ0.56a:
Calculate the mean number of eggs laid by these birds.
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.
21M.1.AHL.TZ2.9a:
Find an estimate for how many copies the vendor expects to sell each day.
21M.1.AHL.TZ2.9b.ii:
Write down the degrees of freedom for this test.
21M.1.AHL.TZ2.9b.iii:
Write down the conclusion to the test. Give a reason for your answer.
17M.2.SL.TZ2.T_1a.ii:
If the teacher chooses a response at random, find the probability that it is a satisfactory response to the Calculus question;
21M.1.AHL.TZ2.9b.i:
State the null and alternative hypotheses for this test.
EXM.1.AHL.TZ0.55a.i:
Calculate the mean number of brown eggs in a box.
EXM.1.SL.TZ0.7b:
State the hypotheses for this t-test.
21M.1.SL.TZ2.8a:
Write down the null and alternative hypotheses.
18M.2.SL.TZ2.T_3e:
State a conclusion for this test. Justify your answer.
SPM.3.AHL.TZ0.1f.i:
Perform a test on the data from school A to show it is reasonable to assume a linear relationship between effort scores and improvements in IB points. You may assume effort scores follow a normal distribution.
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.
21M.2.SL.TZ1.4e:
State the conclusion of the test. Give a reason for your answer.
21M.3.AHL.TZ1.2a:
Use an appropriate test, at the $5 %$ significance level, to determine whether a new employee staying with the firm is independent of their interview rating. State the null and alternative hypotheses, the $p$ -value and the conclusion of the test.
EXN.2.SL.TZ0.4c.ii:
Write down the null and alternative hypotheses.
EXN.2.SL.TZ0.4c.iv:
Determine the conclusion of the test, clearly justifying your answer.
21N.2.SL.TZ0.1a.i:
prefers a tablet.
21N.2.SL.TZ0.1a.ii:
is $11 – 13$ years old and prefers a mobile phone.
21N.2.SL.TZ0.1a.iii:
prefers a laptop given that they are $17 – 18$ years old.
21N.2.SL.TZ0.1a.iv:
prefers a tablet or is $14 – 16$ years old.
21N.2.SL.TZ0.1c:
Write down the number of degrees of freedom.
21N.2.SL.TZ0.1d.i:
Write down the $χ^{2}$ test statistic.
21N.2.SL.TZ0.1d.ii:
Write down the $p$ -value.
21N.2.SL.TZ0.1d.iii:
State the conclusion for the test in context. Give a reason for your answer.
21N.2.AHL.TZ0.7a.i:
Write down null and alternative hypotheses for Loreto’s test.
21N.2.AHL.TZ0.7a.ii:
Using the data from Loreto’s sample, perform the hypothesis test at a $5 %$ significance level to determine if Loreto should employ extra staff.
21N.2.AHL.TZ0.7b.i:
Write down null and alternative hypotheses for this test.
21N.2.AHL.TZ0.7b.ii:
Perform the test, clearly stating the conclusion in context.

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