Solved: The masses in kilograms of melons produced by a farm can be modelled by a... [IB Mathematics Applications & Interpretation (Math AI)]

The masses in kilograms of melons produced by a farm can be modelled by a normaldistribution with a mean of 2.6 kg and a standard deviation of 0.5 kg.Find the probability that two melons picked at random and independently of eachother willOne year due to favourable weather conditions it is thought that the mean mass of themelons has increased.The owner of the farm decides to take a random sample of 16 melons to test this hypothesis at the 5% significance level, assuming the standard deviation of the masses ofthe melons has not changed.Unknown to the farmer the favourable weather conditions have led to all the melonshaving 10% greater mass than the model described above.

Question

HLPaper 2

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

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

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

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

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

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

[2]

Verified

Solution

* 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 $X$ represent the mass of a melon

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

[2 marks]

both have a mass greater than $3.0 kg$ .

[2]

Verified

Solution

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

$= 0.0449 (0.04488 \dots)$ A1

[2 marks]

have a total mass greater than $6.0 kg$ .

[2]

Verified

Solution

Let $T$ represent the total mass

$E (T) = 5.2$ A1

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

$T ~ N (5.2, 0.5)$

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

[4 marks]

Write down the null and alternative hypotheses for the test.

[1]

Verified

Solution

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

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

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

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

[1 mark]

Find the critical region for this test.

[4]

Verified

Solution

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

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

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

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

[4 marks]

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

[3]

Verified

Solution

Let $W$ represent the new mass of the melons

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

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

$= 0.55$ A1

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

[3 marks]

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

[2]

Verified

Solution

$P (Type II error) =$

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

$= 0.346 (0.346204 \dots)$ A1

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

[2 marks]

Question

HLPaper 2

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

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

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

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

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

[2]

Verified

Solution

Let $X$ represent the mass of a melon

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

[2 marks]

both have a mass greater than $3.0 kg$ .

[2]

Verified

Solution

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

$= 0.0449 (0.04488 \dots)$ A1

[2 marks]

have a total mass greater than $6.0 kg$ .

[2]

Verified

Solution

Let $T$ represent the total mass

$E (T) = 5.2$ A1

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

$T ~ N (5.2, 0.5)$

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

[4 marks]

Write down the null and alternative hypotheses for the test.

[1]

Verified

Solution

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

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

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

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

[1 mark]

Find the critical region for this test.

[4]

Verified

Solution

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

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

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

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

[4 marks]

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

[3]

Verified

Solution

Let $W$ represent the new mass of the melons

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

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

$= 0.55$ A1

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

[3 marks]

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

[2]

Verified

Solution

$P (Type II error) =$

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

$= 0.346 (0.346204 \dots)$ A1

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

[2 marks]

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