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Date	November Example question	Marks available	2	Reference code	EXN.2.AHL.TZ0.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	6	Adapted from	N/A

Question

The masses in kilograms of melons produced by a farm can be modelled by a normal distribution with a mean of $2.6 kg$ $2.6 kg$ and a standard deviation of $0.5 kg$ $0.5 kg$ .

Find the probability that two melons picked at random and independently of each other will

One year due to favourable weather conditions it is thought that the mean mass of the melons has increased.

The owner of the farm decides to take a random sample of $16$ $16$ melons to test this hypothesis at the $5 %$ $5 %$ significance level, assuming the standard deviation of the masses of the melons has not changed.

Unknown to the farmer the favourable weather conditions have led to all the melons having $10 %$ $10 %$ greater mass than the model described above.

Find the probability that a melon selected at random will have a mass greater than $3.0 kg$ $3.0 kg$ .

[2]

both have a mass greater than $3.0 kg$ $3.0 kg$ .

[2]

b.i.

have a total mass greater than $6.0 kg$ $6.0 kg$ .

[2]

b.ii.

Write down the null and alternative hypotheses for the test.

[1]

Find the critical region for this test.

[4]

Find the mean and standard deviation of the mass of the melons for this year.

[3]

Find the probability of a Type II error in the owner’s test.

[2]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

Let $X$ $X$ represent the mass of a melon

$P (X > 3.0) = 0.212 (0.2118 \dots)$ $P (X > 3.0) = 0.212 (0.2118 \dots)$ (M1)A1

[2 marks]

$0.2118 \dots \times 0.2118 \dots$ $0.2118 \dots \times 0.2118 \dots$ (M1)

$= 0.0449 (0.04488 \dots)$ $= 0.0449 (0.04488 \dots)$ A1

[2 marks]

b.i.

Let $T$ $T$ represent the total mass

$E (T) = 5.2$ $E (T) = 5.2$ A1

$Var (T) = 0 . 5^{2} + 0 . 5^{2} = 0.5$ $Var (T) = 0 . 5^{2} + 0 . 5^{2} = 0.5$ (M1)A1

$T ~ N (5.2, 0.5)$ $T ~ N (5.2, 0.5)$

$P (T > 6.0) = 0.129 (0.1289 \dots)$ $P (T > 6.0) = 0.129 (0.1289 \dots)$ A1

[4 marks]

b.ii.

Let $μ$ $μ$ be the mean mass of the melons produced by the farm.

$H_{0} : μ = 2.6 kg$ $H_{0} : μ = 2.6 kg$ , $H_{1} : μ > 2.6 kg$ $H_{1} : μ > 2.6 kg$ only A1

Note: Accept $H_{0} :$ $H_{0} :$ The mean mass of melons produced by the farm is equal to $2.6 kg$ $2.6 kg$
$H_{1} :$ $H_{1} :$ The mean mass of melons produced by the farm is greater than $2.6 kg$ $2.6 kg$

Note: Award A0 if $2.6 kg$ $2.6 kg$ does not appear in the hypothesis.

[1 mark]

Under $H_{0}$ $H_{0}$ $¯ ¯ ¯ X ~ N (2.6, \frac{0 . 5^{2}}{16})$ $\bar{X} ~ N (2.6, \frac{0 . 5^{2}}{16})$ A1

$P (¯ ¯ ¯ X > a) = 0.05$ $P (\bar{X} > a) = 0.05$ (M1)

$a = 2.81 (2.805606 \dots)$ $a = 2.81 (2.805606 \dots)$ (A1)

Critical region is $¯ ¯ ¯ X > 2.81$ $\bar{X} > 2.81$ A1

[4 marks]

Let $W$ $W$ represent the new mass of the melons

$E (W) = 1.1 \times 2.6 = 2.86$ $E (W) = 1.1 \times 2.6 = 2.86$ A1

Standard deviation of $W = 1.1 \times 0.5$ $W = 1.1 \times 0.5$ (M1)

$= 0.55$ $= 0.55$ A1

Note: award M1A0 for $Var (W) = 1 . 1^{2} \times 0 . 5^{2} = 0.3025$ $Var (W) = 1 . 1^{2} \times 0 . 5^{2} = 0.3025$

[3 marks]

$P (Type II error) =$ $P (Type II error) =$

$P (\bar{X} < 2.81 μ = 2.86, σ = \frac{0.55}{4})$ (M1)

$= 0.346 (0.346204 \dots)$ A1

Note: Accept $0.358$ from use of the three‐figure answer to part (d)

[2 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.18—T and Z test, type I and II errors

Show 102 related questions

22M.3.AHL.TZ2.1e.ii:
Hence state the probability of a Type I error for this test.
22M.1.AHL.TZ2.12b:
Find the probability of making a Type II error when weighing a female cuttlefish.
EXM.3.AHL.TZ0.8a.ii:
Find unbiased estimates for the population Variance.
EXM.3.AHL.TZ0.8a.i:
Find unbiased estimates for the population mean.
19M.3.AHL.TZ0.Hsp_3e.ii:
Write down the conclusion reached.
22M.3.AHL.TZ1.1a.iii:
Explain why it would not be appropriate to conduct a hypothesis test on the value of $r$ found in (a)(ii).
22M.1.AHL.TZ1.15c:
Find the probability of a Type II error, if the number of cars now follows a Poisson distribution with a mean of $4.5$ cars per minute.
20N.3.AHL.TZ0.Hsp_3a:
Assuming that the shopkeeper’s claim is correct, find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.
21M.3.AHL.TZ2.1f.ii:
State the null and alternative hypotheses for this test.
22M.1.AHL.TZ2.12a:
Find the probability of making a Type I error when weighing a male cuttlefish.
22M.3.AHL.TZ2.1f:
By considering the claims of both Aimmika and Nichakarn, explain whether the advertising was beneficial to the store.
16N.3.AHL.TZ0.Hsp_2b:
Explain why a normal distribution can be used to give an approximate model for $X$ .
EXM.3.AHL.TZ0.8f:
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.
17M.3.AHL.TZ0.Hsp_1c.ii:
Using a 10% significance level and justifying your answer, state your conclusion in context.
18N.3.AHL.TZ0.Hsp_3b.i:
State the distribution of your test statistic, including the parameter.
EXM.3.AHL.TZ0.8d:
Use the normal distribution model to find the score required to pass.
18N.3.AHL.TZ0.Hsp_3c.iii:
State the p-value and interpret it in this context.
18M.3.AHL.TZ0.Hsp_3c:
Find the probability of making a Type II error.
EXM.3.AHL.TZ0.8c:
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.
SPM.1.AHL.TZ0.16b:
Find the probability of a Type I error.
SPM.3.AHL.TZ0.1c.ii:
Find the standard deviation of the changes.
SPM.3.AHL.TZ0.1h:
If you were to repeat the test performed in part (e) intending to compare the quality of the teaching between the two schools, suggest two ways in which you might choose your sample to improve the validity of the test.
SPM.3.AHL.TZ0.1b:
State why comparing only the final IB points of the students from the two schools would not be a valid test for the effectiveness of the two different teaching methods.
SPM.3.AHL.TZ0.1g:
Use an appropriate test to determine whether showing an improvement is independent of gender.
SPM.3.AHL.TZ0.1e.ii:
State why it was important to test that both sets of points were normally distributed.
EXM.2.AHL.TZ0.26b:
The data from the two samples above are combined to form a single set of data. The following frequency table gives the observed frequencies for the combined sample. The data has been divided into five intervals.

Test, at the 5% level, whether the combined data can be considered to be a sample from a normal population with a mean of 380.
EXM.2.AHL.TZ0.28a.iii:
Calculate an appropriate value of ${\chi ^2}$ and state your conclusion, using a 1% significance level.
EXM.2.AHL.TZ0.28a.i:
State suitable hypotheses for testing this belief.
SPM.3.AHL.TZ0.1a:
Identify a test that might have been used to verify the null hypothesis that the predictions from the standardized test can be modelled by a normal distribution.
17N.3.AHL.TZ0.Hsp_2b.i:
Use a two-tailed test to determine the $p$ -value for the above results.
19N.3.AHL.TZ0.Hsp_3c.iii:
Given that the probability that Josie makes a Type II error is 0.25, find the value of $\mu$ , giving your answer correct to three significant figures.
SPM.1.AHL.TZ0.16c:
The average number of fish caught in an hour is actually 2.5.

Find the probability of a Type II error.
19N.3.AHL.TZ0.Hsp_3c.i:
Find the critical region for Josie’s test, giving your answer correct to two decimal places.
17N.3.AHL.TZ0.Hsp_4a:
State suitable hypotheses to investigate whether or not $U$ , $V$ are independent.
SPM.3.AHL.TZ0.1c.i:
Find the mean change.
19M.3.AHL.TZ0.Hsp_3d:
Find a 90 % confidence interval for $\mu$ .
18N.3.AHL.TZ0.Hsp_3c.i:
State suitable hypotheses for the test.
17M.3.AHL.TZ0.Hsp_1c.i:
Carry out an appropriate test and state the $p$ -value obtained.
16N.3.AHL.TZ0.Hsp_2d:
Calculate the probability that he makes a Type II error.
19N.3.AHL.TZ0.Hsp_1a.ii:
Carry out a suitable test at the 5 % significance level. With reference to the $p$ -value, state your conclusion in the context of Peter’s claim.
18N.3.AHL.TZ0.Hsp_3d:
Use an appropriate regression line to estimate the weight of a fish with length 360 mm.
19N.3.AHL.TZ0.Hsp_3c.ii:
Write down the probability that Josie makes a Type I error.
SPM.3.AHL.TZ0.1f.ii:
Hence, find the expected improvement between predicted and final points for an increase of one unit in effort grades, giving your answer to one decimal place.
EXM.3.AHL.TZ0.8b:
Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.
SPM.1.AHL.TZ0.16a:
State a suitable null and alternative hypotheses for Tom’s test.
18M.3.AHL.TZ0.Hsp_3d:
Another model of smartphone whose battery life may be assumed to be normally distributed with mean μ hours and standard deviation 1.2 hours is tested. A researcher measures the battery life of six of these smartphones and calculates a confidence interval of [10.2, 11.4] for μ.

Calculate the confidence level of this interval.
18M.3.AHL.TZ0.Hsp_3a:
State suitable hypotheses for a two-tailed test.
21M.3.AHL.TZ2.1d.i:
State why the hypothesis test should be one-tailed.
21M.3.AHL.TZ2.1f.i:
State the name of the test which Juliet should use.
EXM.3.AHL.TZ0.8g.i:
Find the probability of making a Type I error.
19M.3.AHL.TZ0.Hsp_3c:
Find an unbiased estimate for ${\sigma ^2}$ .
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.
EXN.2.AHL.TZ0.6c:
Write down the null and alternative hypotheses for the test.
19N.3.AHL.TZ0.Hsp_1a.i:
State suitable hypotheses ${H_0}$ and ${H_1}$ to test Peter’s claim, using a two-tailed test.
18N.3.AHL.TZ0.Hsp_3b.iii:
State the conclusion of the test, justifying your answer.
SPM.3.AHL.TZ0.1e.i:
Use an appropriate test to determine whether there is evidence, at the 5 % significance level, that the students in school B have improved more than those in school A.
18N.3.AHL.TZ0.Hsp_3c.ii:
Find the product-moment correlation coefficient $r$ .
EXM.2.AHL.TZ0.28b.i:
Find the significance level of this procedure.
19M.3.AHL.TZ0.Hsp_3a:
State the distribution of $\bar X$ giving its mean and variance.
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.
17N.3.AHL.TZ0.Hsp_4b:
Find the least value of $|r|$ for which the test concludes that $\rho \ne 0$ .
EXM.2.AHL.TZ0.28b.ii:
Some time later, the actual value of $\mu$ is 503. Find the probability of a Type II error.
21M.3.AHL.TZ1.2d.ii:
For the data in this table, test the null hypothesis, $H_{0} : ρ = 0$ , against the alternative hypothesis, $H_{1} : ρ > 0$ , at the $5 %$ significance level. You may assume that all the requirements for carrying out the test have been met.
18N.3.AHL.TZ0.Hsp_3b.ii:
Find the p-value for the test.
17M.3.AHL.TZ0.Hsp_1b:
Find unbiased estimates of $\mu$ and ${\sigma ^2}$ .
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.
17N.3.AHL.TZ0.Hsp_2a:
Determine unbiased estimates for $\mu$ and ${\sigma ^2}$ .
19N.3.AHL.TZ0.Hsp_1b:
Peter uses the regression line of $y$ on $x$ as $y = 0.248x + 83.0$ and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.
16N.3.AHL.TZ0.Hsp_2a:
Find

(i) ${\text{E}}(X)$ ;

(ii) ${\text{Var}}(X)$ .
20N.3.AHL.TZ0.Hsp_3b:
Find the least value of $n$ required to ensure that the width of the confidence interval is less than $2 grams$ .
21M.1.AHL.TZ1.11b:
Write down the null and alternative hypotheses.
18M.3.AHL.TZ0.Hsp_3b:
Find the critical region for testing ${\bar b}$ at the 5 % significance level.
17N.3.AHL.TZ0.Hsp_2b.ii:
Interpret your $p$ -value at the 5% level of significance, justifying your conclusion.
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.
EXM.2.AHL.TZ0.28a.ii:
Calculate the mean of these data and hence estimate the value of $p$ .
19M.3.AHL.TZ0.Hsp_3e.i:
Find the $p$ -value for the test.
21M.3.AHL.TZ1.2e.i:
Write down the number of tests they carry out.
21M.3.AHL.TZ1.2e.iii:
The firm obtains a significant result when comparing section $2$ of the written assessment and attribute $X$ . Interpret this result.
21M.3.AHL.TZ1.2d.iii:
Hence comment on the reliability of the written assessment.
17M.3.AHL.TZ0.Hsp_1a:
State suitable hypotheses to test the inspector’s claim.
21M.1.AHL.TZ2.15a:
State the null and alternative hypotheses for the test.
21M.1.AHL.TZ2.15b:
Find the probability that Sheila will make a type I error in her test conclusion.
21M.1.AHL.TZ1.11d:
State the conclusion of the test. Give a reason for your answer.
19M.3.AHL.TZ0.Hsp_3b:
Find an unbiased estimate for $\mu$ .
21M.3.AHL.TZ2.1d.iii:
The critical value for this test, at the $5 %$ significance level, is $0.549$ . Juliet assumes that the population is bivariate normal.

Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.
21M.3.AHL.TZ2.1d.ii:
State the null and alternative hypotheses for this test.
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.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.
20N.3.AHL.TZ0.Hsp_3d:
Given that the weights of the broccoli actually follow a normal distribution with mean $392 grams$ and variance $80 {grams}^{2}$ , find the probability of Anjali making a Type II error.
16N.3.AHL.TZ0.Hsp_2c:
Use this model to find the values of $A$ and $B$ such that ${\text{P}}(A < X < B) = 0.9$ , where $A$ and $B$ are symmetrical about the mean of $X$ .
20N.3.AHL.TZ0.Hsp_3c:
Find the significance level for this test.
21M.1.AHL.TZ1.11c:
Find the $p$ -value for the test.
21M.3.AHL.TZ1.2e.ii:
The tests are performed at the $5 %$ significance level.

Assuming that:
- there is no correlation between the marks in any of the sections and scores in any of the attributes,
- the outcome of each hypothesis test is independent of the outcome of the other hypothesis tests,
find the probability that at least one of the tests will be significant.
21M.3.AHL.TZ2.1f.iii:
Perform the test, using a $5 %$ significance level, and state your conclusion in context.
21M.1.AHL.TZ2.15c:
Sheila finds $126$ coffees were sold during the $5$ -hour period.

State Sheila’s conclusion to the test. Justify your answer.
EXN.2.AHL.TZ0.6d:
Find the critical region for this test.
EXN.2.AHL.TZ0.6e:
Find the mean and standard deviation of the mass of the melons for this year.
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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Topic 4—Statistics and probability

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