A shop sells carrots and broccoli. The weights of carrots can be modelled by a normaldistribution with variance 25grams2 and the weights of broccoli can be modelled by a normaldistribution with variance 80grams2. The shopkeeper claims that the mean weight of carrotsis 130grams and the mean weight of broccoli is 400grams.Dong Wook decides to investigate the shopkeeper’s claim that the mean weight of carrotsis 130grams. 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, μgrams, of the broccoli is less than 400grams. She decidesto perform a hypothesis test, using a random sample of size 8. Her hypotheses areH0:μ=400;H1:μ<400.She decides to reject H0 if the sample mean is less than 395grams.

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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