Elizabeth's Bakery makes brownies. It is known that the mass, $X$ grams, of a brownie may be modelled by a normal distribution. 10% of the brownies have a mass less than 30 grams. 80% of the brownies have a mass greater than 32.5 grams. (a) Find the mean and standard deviation of $X$. (b) (i) Find $P(X eq 35)$ (ii) Find $P(X < 35)$. (c) Brownies are baked in batches of 13. Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams. You may assume that the masses of brownies are independent of each other.

Photo AI

Elizabeth's Bakery makes brownies - AQA - A-Level Maths Pure - Question 17 - 2019 - Paper 3

Question 17

Elizabeth's Bakery makes brownies. It is known that the mass, $X$ grams, of a brownie may be modelled by a normal distribution. 10% of the brownies have a mass les... show full transcript

Worked Solution & Example Answer:Elizabeth's Bakery makes brownies - AQA - A-Level Maths Pure - Question 17 - 2019 - Paper 3

Step 1

Find the mean and standard deviation of $X$.

96%

114 rated

Answer

Given that 10% of the brownies have a mass less than 30 grams, we can find the z-value using the inverse normal distribution for $P(Z < z) = 0.10$ . This gives us:

$z ext{ for } 0.10 = -1.2816$

Using the formula for z:

$z = rac{X - ext{mean}}{ ext{std. dev.}}$

We have:

$-1.2816 = rac{30 - ext{mean}}{ ext{std. dev.}}$

Next, for the 80% statement:

Since 80% have a mass greater than 32.5 grams, 20% have a mass less than or equal to 32.5 grams. Hence, we find:

$P(Z < z) = 0.20$

This gives:

$z ext{ for } 0.20 = -0.8416$

Using the formula:

$-0.8416 = rac{32.5 - ext{mean}}{ ext{std. dev.}}$

Now, we have a system of two equations:

$-1.2816 = rac{30 - ext{mean}}{ ext{std. dev.}}$
$-0.8416 = rac{32.5 - ext{mean}}{ ext{std. dev.}}$

We can solve this system simultaneously to find both the mean and standard deviation. Substituting values into the equations leads to:

$ext{mean} = 37.3 ext{ grams}$ $ext{std. dev.} = 5.68 ext{ grams}$

Step 2

Find $P(X eq 35)$

99%

104 rated

Answer

To find this probability, we use the complement rule:

eq 35) = 1 - P(X = 35)$$ Since we are dealing with a continuous distribution, we have: $$P(X = 35) = 0$$ Thus: $$P(X eq 35) = 1$$

Step 3

Find $P(X < 35)$

96%

101 rated

Answer

First, we need to calculate the z-score for $X = 35$ using the mean and standard deviation found earlier:

$z = rac{35 - 37.3}{5.68} = -0.404$

Using the standard normal distribution table or calculator:

$P(X < 35) ext{ corresponds to } ext{P(Z < -0.404) = 0.344}$

Step 4

Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams.

98%

120 rated

Answer

We model this situation using a Binomial distribution, where:

$n = 13$ (the number of brownies)
$p = P(X < 35) = 0.344$

We want to find:

$P(Y ext{ is no more than } 3) = P(Y ext{ ≤ } 3)$

Using the Binomial probability formula:

$P(Y = k) = inom{n}{k} p^k (1-p)^{n-k}$

Thus:

$P(Y ext{ ≤ } 3) = inom{13}{0} (0.344)^0 (0.656)^{13} + inom{13}{1} (0.344)^1 (0.656)^{12} + inom{13}{2} (0.344)^2 (0.656)^{11} + inom{13}{3} (0.344)^3 (0.656)^{10}$

Calculate these probabilities to find the final answer.

Elizabeth's Bakery makes brownies - AQA - A-Level Maths Pure - Question 17 - 2019 - Paper 3

Worked Solution & Example Answer:Elizabeth's Bakery makes brownies - AQA - A-Level Maths Pure - Question 17 - 2019 - Paper 3

Find the mean and standard deviation of $X$.

Find $P(X eq 35)$

Find $P(X < 35)$

Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams.

Join the A-Level students using SimpleStudy...