Solved: A shop sells carrots and broccoli. The weights of carrots can be modelled by a... [IB Mathematics Applications & Interpretation (Math AI)]

Question

HLPaper 3

A shop sells carrots and broccoli. The weights of carrots can be modelled by a normaldistribution with variance $25 {grams}^{2}$ and the weights of broccoli can be modelled by a normaldistribution with variance $80 {grams}^{2}$ . The shopkeeper claims that the mean weight of carrotsis $130 grams$ and the mean weight of broccoli is $400 grams$ .

Dong Wook decides to investigate the shopkeeper’s claim that the mean weight of carrotsis $130 grams$ . He plans to take a random sample of $n$ carrots in order to calculate a $98 %$ confidence interval for the population mean weight.

Anjali thinks the mean weight, $μ grams$ , of the broccoli is less than $400 grams$ . She decidesto perform a hypothesis test, using a random sample of size $8$ . Her hypotheses are

$H_{0} : μ = 400; H_{1} : μ < 400$ .

She decides to reject $H_{0}$ if the sample mean is less than $395 grams$ .

Assuming that the shopkeeper’s claim is correct, find the probability that the weight ofsix randomly chosen carrots is more than two times the weight of one randomlychosen broccoli.

[6]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

Let $X = Σ_{i = 1}^{6} C_{i} - 2 B$ M1

$E (X) = 6 \times 130 - 2 \times 400 = - 20$ (M1)(A1)

$Var (X) = 6 \times 25 + 4 \times 80 = 470$ (M1)(A1)

$P (X > 0) = 0.178$ A1

Note: Condone the notation $6 C - 2 B$ only if the (M1) is awarded for the variance.

[6 marks]

Find the least value of $n$ required to ensure that the width of the confidence interval isless than $2 grams$ .

[3]

Verified

Solution

$z = 2.326 \dots$ (A1)

$\frac{2 z σ}{\sqrt{n}} < 2$ M1

$\sqrt{n} > 11.6 \dots$

$n > 135.2 \dots$

$n = 136$ A1

Note: Condone the use of equal signs.

[3 marks]

Find the significance level for this test.

[3]

Verified

Solution

variance $= \frac{80}{8} = 10$ (A1)

under $H_{0}, \bar{B} ~ N (400, 10)$

significance level $= P (\bar{B} < 395)$ (M1)

$= 0.0569$ or $5.69 %$ A1

Note: Accept any answer that rounds to $0.057$ or $5.7 %$ .

[3 marks]

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.

[3]

Verified

Solution

Type II error probability $= P (Accept H_{0} H_{1} true)$ (M1)

$= P (\bar{B} > 395 \bar{B} \approx N (392, 10))$ (A1)

$= 0.171$ A1

Note: Accept any answer that rounds to $0.17$ .

[3 marks]

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