A-Level Edexcel Maths Statistics Binomial Distribution: A company sells seeds and claims

A company sells seeds and claims that 55% of its pea seeds germinate - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 2

Question 5

A company sells seeds and claims that 55% of its pea seeds germinate. (a) Write down a reason why the company should not justify their claim by testing all the pea ... show full transcript

Worked Solution & Example Answer:A company sells seeds and claims that 55% of its pea seeds germinate - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 2

Step 1

Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.

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Answer

The seeds would be destroyed in the process so they would have none to sell.

Step 2

Assuming that the company’s claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.

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Answer

Let S = no. of seeds out of 24 that germinate, S ~ B(24, 0.55).

Let T = no. of trays with at least 15 germinating, T ~ B(10, p) where p = P(S ≥ 15).

First, calculate P(S ≥ 15):

$P(S ext{ } ext{ if } ext{ } S = 15) ext{ } ... ext{ } ext{ } P(S >= 15) ext{ } = 0.1487$

Then, for T:

$P(T ext{ } ext{ if } ext{ } T >= 5) ext{ } = 0.149$

Step 3

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution.

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Answer

n is large and p is close to 0.5.
The sample size n must satisfy both np and n(1-p) > 5.

Step 4

Assuming that the company’s claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.

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Answer

Let X ~ N(132, 59.4^2).

Calculate:

$P(X > 149.5) = P \left( Z > \frac{149.5 - 132}{59.4} \right)$

Using continuity correction gives:

$P(Z > 0.293... ) = 0.01158... ext{ which approximates to } 0.0116$

Step 5

Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of 55%.

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Answer

The probability is very small; therefore, there is evidence that the company’s claim is incorrect.

A company sells seeds and claims that 55% of its pea seeds germinate - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 2

Worked Solution & Example Answer:A company sells seeds and claims that 55% of its pea seeds germinate - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 2

Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.

Assuming that the company’s claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution.

Assuming that the company’s claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.

Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of 55%.

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