S5.1 Hypothesis-test language; correlation tests

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Examiner's comment

In (b)(i) the hypotheses were usually correctly stated with most students using a one-tailed test. On some occasions students are writing these hypotheses in words rather than using appropriate notation. Most students were able to select and use the correct Binomial distribution. Many, however, then calculated P(X = 11) or incorrectly attempted P(X >= 11) = 1 – P(X <= 11). Of the students who gained the A mark for the probability 0.16, the conclusion was usually correct and mostly given in sufficient contextual detail. Those students who used a critical region approach generally did less well, as most omitted the required p-value and scored a maximum of 3 marks. relevant probabilities at some point in part (b). A few students gave a probability rather than a region as their final answer here.

Common mistakes

• H0: p = 0.2, H1: p > 0.2 in terms of p (not words)
• X ~ B(40, 0.2)
• p-value P(X ≥ 11) = 1 − P(X ≤ 10) = 0.161 (not P(X = 11) and not 1 − P(X ≤ 11))
• do not reject H0, insufficient evidence for Jayda's belief, in context
• a critical-region approach without the p-value scores at most 3

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Examiner's comment

In (b)(ii) students appeared unfamiliar with the vocabulary ‘acceptance region’. A very common mistake was an attempt to find the critical region. It was not uncommon to see P(X <= 12 ) = 0.9568 followed by an acceptance region of X >= 13. Those who did realise that the acceptance region was not the same as the critical region often reverted to a 2-tailed approach. Students were able to earn the method mark here by calculating one or more relevant probabilities at some point in part (b). A few students gave a probability rather than a

Common mistakes

• acceptance region is the complement of the critical region: P(X ≤ 12) = 0.9568 > 0.95 so 0 ≤ X ≤ 12
• common errors: giving the critical region X ≥ 13, switching to a two-tailed test, giving a probability statement instead of a region

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Examiner's comment

In part (b) most responses had the correct binomial model or inferred it by stating a suitable probability. A common mistake though was to use P(X = 8) rather than 𝑃𝑃(𝑋𝑋⩾8) which did not gain either of the remaining marks. Those with the correct probability sometimes gave a comparison with 0.05 rather than 0.025 and did not gain the final mark and others did not give a conclusion in context. Although a critical region was not required here (we will only ask for this if it can be found using the tables provided) some did use this approach. Many were successful but a few did not give both tails of the critical region and others did not state the critical region properly for example by just stating ( ) P 9 0.0177... X = and not writing the critical region as 𝑋𝑋⩾9 and X = 0. A small number attempted to use a normal approximation here.

Common mistakes

• X ~ B(260, 0.015)
• use P(X ≥ 8) = 0.0441, not P(X = 8)
• two-tailed so compare with 0.025, not 0.05
• conclusion in context (insufficient evidence the proportion of defective components has changed)
• a critical region needs both tails

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Examiner's comment

learners and very few blank responses were seen and parts (e) and (f) provided an opportunity for differentiation. Part (e) was found to be much more challenging for learners. Many were able to state the hypotheses correctly although some had an alternative hypothesis of ρ >0 and a few had a null hypothesis of ρ = 0.015 though few were using r instead of ρ. Very few responses interpreted the p-value with many comparing it to 0.16 or comparing 0.16 with 0.05 but some gave a correct conclusion in context. Good attempts to part (f) were rare and blank responses

Common mistakes

• H₀: ρ = 0, H₁: ρ < 0 (not ρ > 0, not r, not 0.015)
• compare the p-value 0.015 with 0.05 (not −0.160)
• conclude in context: evidence to support the teacher's claim

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Examiner's comment

For those who attempted part (b), this was answered much better than in previous series and students are becoming more familiar with the structure of these questions, writing the hypotheses in terms of p and the distribution clearly. Sometimes the probability statements were not clearly stated, with students writing down the values that they had inputted into their calculators. It was quite common to see P(𝑋 < 3) or P(𝑋 = 3) rather than P(𝑋 <= 3) calculated. The conclusion was not always given in context. When the context was given, it was usually correct, although missing out the word “proportion” or its equivalent meant the mark was lost. Some students chose to use a critical region approach rather than a p-value approach, but they often did not make this clear and some were then unsure about whether 3 was inside or outside the critical region and what this implied for the acceptance or otherwise of H0 Part (d) was frequently omitted or the value p = 0.12 given. Those who used the critical region approach rarely got this mark. However, students are becoming more aware of what the p-value represents.

Common mistakes

• H₀: p = 0.12, H₁: p < 0.12 (in terms of p)
• D ~ B(60, 0.12)
• P(D ⩽ 3) = 0.0601 > 0.05 (not P(D < 3) or P(D = 3))
• do not reject H₀: insufficient evidence that the proportion of defective items has decreased
• conclusion in context with "proportion"
• CR approach must be made clear (CR D ⩽ 2)

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Examiner's comment

This question was generally well attempted, but there are still a significant number of students who show no understanding of hypothesis testing. Some students truncated the values seen on their calculators rather than giving them to an appropriate degree of accuracy.

Common mistakes

• p-value = 0.06 (the probability P(D ⩽ 3) from (b))
• giving 0.12 or the CR probability 0.0196 is wrong
• often omitted

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Examiner's comment

Whilst most students showed they had a good grasp of the concepts of correlation and flight. In part (b) most students interpreted this situation as a one-tailed test and stated the hypotheses correctly in terms of but a few equated this equal to – 0.510 rather than 0 for the null hypothesis. Almost all the students gave a critical value that matched their alternative hypothesis, and an encouraging number gave a correct contextual conclusion to the test mentioning both height and time. Some however lost the final accuracy mark for having contradictory or incorrect statements such as “reject H0” followed by a correct contextual conclusion such as “there is insufficient evidence of a negative correlation between h and t”. Although we did not require a comparison of the test statistic with the critical value if one was seen it had to be correct and some students lost the final mark for statements such as – 0.510 < – 0.5494 or for comparing 0.510 with the significance level of 0.05 In part (c) most

Common mistakes

• H₀: ρ = 0, H₁: ρ < 0 (not ρ = −0.510, not r)
• one-tail cv 0.5494
• not significant: insufficient evidence of negative correlation between height and time
• no contradictory "reject H₀"
• do not compare r with 0.05

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Examiner's comment

In part (a) most gave the hypotheses correctly in terms of p though some failed to have the H0 and H1 labels and some had p = 0 for the null hypothesis perhaps confusing this with a test for correlation. Most used the correct distribution to try and find the critical region but notation was often poor with P(X = 8) being written when P(X ⩽ 8) was meant or calculated and a number struggled to find P(X ⩾ 9) correctly leading to upper tails of the critical region of X ⩾ 8 or X ⩾ 10. There were many errors with the probabilities in each tail with P(X = 0) = 0.0147 (a truncated answer) frequently seen instead of 0.0148 the correct rounded value. Students don’t seem to be aware that the table function on their calculator truncates values rather than rounding them. Some students gave their critical regions inside probability statements and of course failed to secure the accuracy mark, and some didn’t think that X = 0 was a valid part of the critical region and omitted this from their answer. Many students who had identified a critical region were able to add their tail probabilities to give the significance level, but some merely wrote down 0.025 or doubled the probability of 0.0155 found in part (a). this against their probabilities of 0.0148 or 0.0155 from part (a).

Common mistakes

• H₀: p = 0.1, H₁: p ≠ 0.1 (two-tail, not p = 0)
• X ~ B(40, 0.1)
• P(X = 0) = 0.0148 (calculator table truncates to 0.0147), P(X ≥ 9) = 1 − P(X ≤ 8) = 0.0155
• CR X = 0 or X ≥ 9, not inside probability statements and not omitting X = 0

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (a) most gave the hypotheses correctly in terms of p though some failed to have the H0 and H1 labels and some had p = 0 for the null hypothesis perhaps confusing this with a test for correlation. Most used the correct distribution to try and find the critical region but notation was often poor with P(X = 8) being written when P(X ⩽ 8) was meant or calculated and a number struggled to find P(X ⩾ 9) correctly leading to upper tails of the critical region of X ⩾ 8 or X ⩾ 10. There were many errors with the probabilities in each tail with P(X = 0) = 0.0147 (a truncated answer) frequently seen instead of 0.0148 the correct rounded value. Students don’t seem to be aware that the table function on their calculator truncates values rather than rounding them. Some students gave their critical regions inside probability statements and of course failed to secure the accuracy mark, and some didn’t think that X = 0 was a valid part of the critical region and omitted this from their answer. Many students who had identified a critical region were able to add their tail probabilities to give the significance level, but some merely wrote down 0.025 or doubled the probability of 0.0155 found in part (a). In part (c) some students didn’t have a critical region of the correct form and were unable to access this mark. Others simply said that they would “accept H0” without any attempt to give a contextual conclusion and some thought that because the test was not significant Freya’s belief was therefore correct. A few students incorrectly used the probability 7 and then compared 40 this against their probabilities of 0.0148 or 0.0155 from part (a).

Common mistakes

• actual significance level 0.0148 + 0.0155 = 0.0303
• not 0.025 or double one tail

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Examiner's comment

There were some good attempts seen at this question with most students able to display some knowledge of hypothesis testing. Part (a) was very well answered with most students scoring this mark for a correct alternative hypothesis using the correct notation. A few students did not appreciate that this was a two- tailed test and gave a one-tailed alternative hypothesis.

Common mistakes

• two-tailed: H1: p ≠ 0.25 in terms of p
• some gave a one-tailed H1

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Examiner's comment

There were some good attempts seen at this question with most students able to display some knowledge of hypothesis testing. In part (b), when attempted, most students acknowledged that it was a Binomial distribution and many were able to find the probability associated with at least one of the tails (usually the lower one). However, some did not take notice in the question that the probability in each tail was required to be as close to 2.5% as possible, and hence found an incorrect upper critical region. Marks were sometimes lost for failing to state the probability associated with each tail (particularly the upper tail) and for incorrectly identifying the upper critical region, usually opting for X >= 20. There are a significant number of students writing the critical region as a probability. Many students with an incorrect critical region were still able to earn the mark in

Common mistakes

• X ~ B(50, 0.25)
• P(X ≤ 6) = 0.0194 and P(X ≥ 19) = 0.0287 must both be stated
• CR X ≤ 6 or X ≥ 19 (X ≥ 20 common error)
• point probabilities such as P(X = 6) are not tail probabilities
• CR written as a probability

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Examiner's comment

There were some good attempts seen at this question with most students able to display some knowledge of hypothesis testing. part (c) using the correct probabilities for their tails.

Common mistakes

• actual significance level = sum of the two tail probabilities = 0.048 (ft their CR)

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Examiner's comment

This question was also answered well. The hypothesis test in part (d) was answered well with correct notation used for the hypotheses and the appropriate one-tailed test identified. Most selected the correct binomial model but it was disappointing to see a number finding P(X = 9) rather than P(X ⩽ 9). A correct conclusion in context was often given: some using the simple response that there is evidence to support Kamil’s claim and others stating that there was 1 evidence that the proportion of packets containing a prize is less than 7. It was encouraging to see that most of those using the second approach were referring to “proportion” rather than “number of” this time.

Common mistakes

• H0: p = 1/7, H1: p < 1/7 (one-tailed)
• use P(X ≤ 9) = 0.0383, not P(X = 9)
• conclusion in context about the proportion, not the number, of packets

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Examiner's comment

In part (b) most students stated their hypotheses correctly in terms of with very few using a one-tailed test. To score the method mark in this type of question the students simply need to select and write down the correct model (distribution). A number seemed to be listing numbers they needed for their calculator, but we expect to see the appropriate mathematical notation used. Fortunately, many obtained a correct probability and would therefore receive the method mark by implication. Some chose to standardise and find a z value (1.9088… which should be compared with 1.96) or a critical region

Common mistakes

• two-tailed: H1: μ ≠ 175.4, compare 0.028 with 0.025 (or z = 1.909 with 1.96)
• sample mean ~ N(175.4, 6.8²/52) under H0, not mean 177.2
• conclusion in context: insufficient evidence of a difference

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Examiner's comment

(𝑆̄ ⩾ 177.248. . . ) but given that a p-value was required in part (c) this was not the most efficient route. A common error in stating the model is to use an incorrect value for the 2 6.8 mean. On this occasion we condoned for the M1 and first A1 but ~ N 177.2, S 52 penalised the final accuracy mark. In a hypothesis test the probability (or critical region) is always calculated assuming H0 is true and so the mean should be 175.4 and the probability calculated is ( ) S Some students stopped after stating that the result P 177.2 . was not significant or not rejecting H0 but a good number gave a correct conclusion in context. Part (c) was not answered well with only a few scoring this mark. The common error was to use 0.028 or 0.025 but others used values like 175.4

Common mistakes

• two-tailed p-value = 2 × 0.0281 = 0.056
• common errors 0.028 or 0.025

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Examiner's comment

about which function to use on their calculator and ending up with P(D = 3). In part (b) the hypotheses were usually correctly stated using p and a correct model selected. Most obtained the correct probability although some found P(X = 2) rather than P(X „ 2). The correct decision was usually made but some failed to give a suitable contextual conclusion: the simplest description mentioned that there was evidence to support supplier B’s claim but other chose to talk about damp bags and often referred to the “number of bags” rather than the “proportion” decreasing. This type of question is now fairly familiar on these papers but there were a number of students who did not seem to have covered the topic of hypothesis testing and nearly 15% scored zero on this question.

Common mistakes

• H0: p = 0.08, H1: p < 0.08 in terms of p
• X ~ B(70, 0.08)
• P(X ≤ 2) = 0.074 (not P(X = 2)) < 0.10 so reject H0
• conclusion in context about the proportion (not the number) of damp bags, supporting B's claim

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Examiner's comment

Part (a) was well answered with the majority of students writing their hypotheses in terms of p.

Common mistakes

• H0: p = 0.1, H1: p ≠ 0.1 in terms of p (two-tailed for "a change")

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Examiner's comment

Part (b) proved to be a challenge for many of the students with a reasonable proportion unable to even begin, which is unusual for a question of this nature – as it is quite standard. Most students were able to state the correct distribution and write some probabilities, but many were not able to identify correctly the CR from their probabilities. Responses scoring M1A1A1 were rare. This was often due students giving probability statements rather than giving a CR Other issues included: • giving probabilities to only 2 s.f. • selecting the region between the two Critical values as their CR Students who were aware of what a two-tailed CR looks like almost always got part (c) For those students who stated both tails of the critical region in (b), the majority correctly compared 15 with their critical region and then usually supported the manager. Others used the longer version of the contextual statement and most did so correctly.

Common mistakes

• B(50, 0.1)
• critical region X = 0 or X ≥ 10 with P(X = 0) = 0.0052 and P(X ≥ 10) = 0.0245
• give regions not probability statements
• not the region between the critical values

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Examiner's comment

Students who were aware of what a two-tailed CR looks like almost always got part (c) correct. The students who did not really understand the idea of the correct CR seemed to make a fairly random guess at the actual significance level. For those students who stated both tails of the critical region in (b), the majority

Common mistakes

• 0.0052 + 0.0245 = 0.0297
• add both tails of the critical region

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Examiner's comment

Whilst many students realised that one- tailed hypotheses were needed in part (b) few used ρ for the parameter. Most students had ± the correct CV but the final A1 was lost by many students, as they compared the positive value 0.3887 to the negative test statistic. This showed these students were unfamiliar with the meaning of the CV/CR and how it related to their test statistic. Part (d) differentiated well between students with basic knowledge and those with deeper understanding. The majority of students scored 5/5 on this part. Almost all used the LH approach on the MS. This was usually well done, though a good number made errors with the laws of indices and maintained a + between the two terms after making the equation into powers of 10. The students knew their log rules, but many did only so to a basic degree, and came unstuck during their working. Sometimes this was to such an extent that they lost the marks. A surprisingly high number of students used natural logarithms and hence the exponential function in this question showing a less secure grip of this topic.

Common mistakes

• H0: ρ = 0, H1: ρ < 0 (ρ, one-tailed)
• critical value −0.3887
• −0.897 < −0.3887 so significant: evidence of negative correlation
• comparing 0.3887 with −0.897 the wrong way

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Examiner's comment

Part (d) was a standard binomial hypothesis test and most students stated correct hypotheses and a correct model. There was still some confusion about the appropriate probability to calculate and a significant number found P(X = 4) rather than P( X „ 4). Those who did find the correct probability were usually able to give a correct conclusion in context which is encouraging however few knew what the p- value was in part (e).

Common mistakes

• two-tailed: H₀ p = 0.14, H₁ p ≠ 0.14
• X ~ B(75, 0.14)
• P(X ≤ 4) = 0.0151 compared with 0.025, not P(X = 4)
• conclusion in context: evidence the proportion of red beads has changed

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Examiner's comment

value was in part (e).

Common mistakes

• two-tailed test so p-value = 2 × 0.0151 = 0.03
• few knew what the p-value was

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Examiner's comment

Most answered part (c) correctly though a number “lost” the minus sign between their calculator and writing down the answer on the page. Usually there were sufficient figures given for us to award the mark when the minus sign was included. In part (d) most attempted the hypotheses and the majority used ρ. Some

Common mistakes

• calculator r = −0.545
• do not lose the minus sign

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Examiner's comment

by the scatter diagram and did not reconsider this even after correctly answering part (d). Most answered part (c) correctly though a number “lost” the minus sign between their calculator and when the minus sign was included. In part (d) most attempted the hypotheses and the majority used ρ. Some didn’t have “= 0” for their null hypothesis with alternative values or inequalities being used and of course some used r or simply wrote the hypotheses in words. We allowed (+) for the critical value provided the actual value was compatible with their alternative hypothesis and many achieved this mark but few scored the final mark as the conclusion was not related to the context of the question.

Common mistakes

• H₀: ρ = 0, H₁: ρ < 0 in terms of ρ (not r or words)
• 1-tail 5% critical value −0.4259
• conclusion in context (correlation between numbers of letters in last and first names)

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Examiner's comment

This question required the binomial distribution and it was pleasing to see that around 65% of candidates were able to recognise the distribution required in Again in part (c) many candidates gave the correct model, but fewer were able to write down the hypotheses correctly. Of those who progressed to the end, most evaluated the required probability, but a not insubstantial minority worked out the probability of exactly eleven. The great majority of those who found the correct probability then went on to give the correct answer in context.

Common mistakes

• H0: p = 1/6, H1: p > 1/6 in terms of p
• R ~ B(35, 1/6)
• P(R ≥ 11) = 0.0232 < 0.05 (not P(R = 11))
• conclusion in context supporting Rowan's belief, "reject H0" alone not enough

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Examiner's comment

not available on the large data set for Perth. Part (c) was a standard hypothesis test for correlation. It was encouraging to see a good number using ρ for the hypotheses but fewer realised that the test was two-tailed. The phrase “…believes there is a correlation between…” shows that a two tailed test should be used, however many were swayed by the negative value of r and opted for a one tailed test. It was good to see that most chose the critical value from the tables that was compatible with their alternative hypothesis and this meant that they could also access the final mark provided their comment was in context, “support for Stav’s belief” was perhaps the simplest way of including the context. There were some good answers to part (d) but many others were

Common mistakes

• "a correlation" means a two-tailed test: H₀: ρ = 0, H₁: ρ ≠ 0 (many chose one tail because r < 0)
• critical value −0.3610
• conclusion in context (support for Stav's belief)

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Examiner's comment

including the context. There were some good answers to part (d) but many others were unable to pull together the various parts of the question to make a correct inference. Some stated that the humidity was high and other assumed this but interpreted the negative value of r correctly and were then able to state that they would expect the number of hours of sunshine to be low. Some gave a suggested value for the number of hours of sunshine but this needed to be realistic and values more than 5 were not accepted. A few students knew (from the large data set) that humidity greater than 95% implies foggy conditions and argued correctly that this would suggest a low amount of sunshine.

Common mistakes

• high humidity with negative correlation (or fog above 95% humidity) means low sunshine, below the Heathrow average
• a suggested value must be realistic (5 hours or less)

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Examiner's comment

Only the most able candidates on the paper made significant progress through this question on hypothesis testing. In part (a) the first error in writing down the alternative hypothesis was identified and corrected more often than the error in the probability statement. Some candidates lost the marks by identifying where the error was but without correctly identifying what the error was. It was not uncommon for candidates to say that the probability distribution was used rather than the cumulative distribution; likely referring to calculator functions rather than demonstrating a formal understanding of hypothesis testing.

Common mistakes

• H₁ should be p > 0.15 (not ≥)
• the test statistic should be P(X ≥ 8) = 0.0698, not P(X = 8)
• say what the correct version is, not just where the error is

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Examiner's comment

Only the most able candidates on the paper made significant progress through this question Despite the question stating the test should be one-tailed in part (c) some candidates found a two-tail critical region or went for the left hand tail rather than the right hand one. Where the upper tail was found the critical region was usually correct although some gave a critical value rather than a region or wrote it as a probability i.e. P(X >= 9). Some candidates rewrote Julia’s hypothesis test correctly at this point, not appreciating the difference between this and a critical region approach. Part (d) was usually correctly done by those who were successful in part (c). On some occasions answers were given only to two significant figures. Just over 5% of candidates scored full marks on the final question on this paper.

Common mistakes

• one-tailed upper tail of B(30, 0.15): P(X ≥ 9) = 0.0277 < 0.05, P(X ≥ 8) > 0.05
• CR X ≥ 9 (not a two-tailed or lower tail region, not P(X ≥ 9) as a probability, not just the critical value)

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Examiner's comment

Only the most able candidates on the paper made significant progress through this question Part (d) was usually correctly done by those who were successful in part (c). On some occasions answers were given only to two significant figures. Just over 5% of candidates scored full marks on the final question on this paper.

Common mistakes

• actual significance level = P(X ≥ 9) = 0.0278 to 3 s.f.

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Examiner's comment

This was the second most successful question on the paper with nearly a quarter of candidates scoring full marks. In part (a) the required symbol ρ was often written as p in the hypotheses and though this was condoned for the first mark. Candidates who lost the mark were mainly using r or pmcc instead of the parameter ρ. Occasional two-tailed hypotheses were stated. The majority of candidates were successful in finding the critical value but some final conclusions were confused with candidates thinking that the critical region was where the pmcc was less than the critical value. A small number of candidates incorrectly tried to apply a Binomial distribution writing X~B(24, 0.446) or tried to standardise using 1.6449 = (x – 24)/0.446 to find the critical value. Many gave a correct response in part (b) using the idea that the correlation was stronger or the

Common mistakes

• H0: ρ = 0, H1: ρ > 0 (ρ, not r or pmcc)
• critical value 0.3438
• 0.446 > 0.3438 so significant: evidence of positive correlation
• not a binomial test

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Examiner's comment

This was the second most successful question on the paper with nearly a quarter of candidates Many gave a correct response in part (b) using the idea that the correlation was stronger or the value closer to 1. Some candidates only compared the value of 0.822 to 0.446 (or 0 or 0.3438) without explaining how this supported Barbara’s belief by showing a stronger correlation compared to the uncoded data. A common incorrect answer was to state that 0.882 was close to 0.89, the gradient of the regression equation. A variety of approaches were taken to answer part (c). Those that opted to take method 2

Common mistakes

• 0.882 is closer to 1 than 0.446 – stronger correlation for the log-coded data
• just comparing with 0.3438 or saying 0.882 is close to the gradient 0.89 scores nothing
• "association" not allowed

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Examiner's comment

In the new specification, students are required to evaluate binomial probabilities using their using a formula, part (ii) was not feasible that way . In the hypothesis test in part (b), the hypotheses were usually given in terms of p and most identified the correct model to use B(32, 1 3 ). A correct probability calculation did not always follow: some found P(X = 16) instead of P(X … 16) and we would usually expect values to be given to 3 significant figures, but some students gave no value at all simply stating that the probability was < 0.05. In order to award the mark we do need sufficient evidence that the correct value has been obtained. The final mark of a hypotheses test will always require a correct conclusion in context: in this case a statement saying there is evidence to support Naasir’s claim is all that we expected.

Common mistakes

• H0: p = 1/3, H1: p > 1/3 (not p ≥ 1/3)
• X ~ B(32, 1/3)
• P(X ≥ 16) = 0.0377 < 0.05, not P(X = 16)
• give the probability value, not just "< 0.05"
• conclusion in context: evidence to support Naasir's claim

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Examiner's comment

This question was answered quite well by most students. In part (a), some failed to give their hypotheses in terms of a population parameter but most quoted the correct critical value from the tables. We would always expect a conclusion to a hypothesis test to be given in context and many students failed to do this.

Common mistakes

• hypotheses in terms of rho (H0: rho = 0, H1: rho < 0), not r
• one-tail critical value −0.6215
• conclusion in context mentioning negative correlation between w and t

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Examiner's comment

and temperature. Part (c) was answered very well, but in part (d) some students seemed unfamiliar with the term, all that was required was the identification of temperature as the required variable and an explanation that, in this model, sales depend on or are affected by temperature. Part (e) was

Common mistakes

• r close to −1 (strong/near perfect linear correlation) so consistent with a linear model