[Maximum mark: 6]

Consider the following normal distributions.

a) Normal distribution with \( \mu = 5,000 \) and \( \sigma = 250 \). What proportion of the data falls between \( 4,750 \) and \( 5,250 \)? Between \( 4,500 \) and \( 5,500 \)?

b) Normal distribution with \( \mu = 375 \) and variance \( = 225 \). What proportion of the data falls between \( 350 \) and \( 400 \)? Between \( 300 \) and \( 450 \)?

[Maximum mark: 8]

A family is planning their summer holiday. They want to go to the beach each time the weather permits. The probability distribution of clear and sunny days is as follows:

x	6	7	8	9	10	11	12	13	14
P(x)	0.04	0.07	0.20	0.22	0.19	0.12	0.08	0.06	0.02

What is the probability that the family will go to the beach:

a) At least 11 times?

b) Fewer than 8 times?

c) Between 8 and 10 times?

d) More than 14 times?

[Maximum mark: 6]

The bags of potatoes are being packed in a store. Their weight (\( W \)), follows a normal distribution with a mean of 800g, and a standard devation of 50g.

A bag of potatoes is selected at random.

a) Find the probability that it weighs less than 700g.

b) In a random selection of 20 bags of potatoes, find the probability that exactly 1 of them wieghs less than 700g.

[Maximum mark: 10]

The savings of people in a city are normally distributed with a mean of $4,000 and a standard deviation of $1,200.

a) If we take two people at random in this city, what is the probability that they both have savings between $2,800 and $5,200?

b) How much money do you need to have to be in the top 10% of wealth in this city?

c) 75% of people have less than \( d \) dollars. Find \( d \).

d) What proportion of people have less than $3,000?

[Maximum mark: 12]

A farm produces eggs and has found that their weights follow a normal distribution with mean 48.8g and standard deviation 3.40g

An egg is considered small if its weight is in the lighter 25%, and it is considered big if it is heavier than 75% of them.

a) Sketch a diagram of this weight distribution.

b) What is the maximum weight of a small egg?

c) What is the minimum weight of a big egg?

We take 50 eggs at random from this farm. Determine the expected number of eggs that:

d) Weigh less than 46g.

e) Weigh between 45g and 55g.

[Maximum mark: 9]

The dimater of watermelons in a particular shop are normally distributed with a mean 12cm with a standard deviation of 1.2cm

A watermelon is chosen at random.

a) Find the probability that:

i. the watermelon has a diameter less than 10cm.

ii. the watermelon has a diameter more than 15cm.

20% of watermelons are known to have a diameter bigger than \( d \).

b) Find the value of \( d \). Round your answer to the nearest integer.

Watermelons with a diameter smaller than 8.5cm are discarded by the shop due to the lack of demand for them.

c) Find the probability that a randomly selected watermelon is discarded due to the lack of demand.

[Maximum mark: 9]

Cars in a given car company have various prices with a mean of $25,000 and standard deviation of $3,500.

The car dealership wants to put them into categories based on their prices, where they will be classified as cheap, standard, and expensive. Expensive cars have the price range between $40,000 and $48,000.

a) Find the probability that a randomly selected car will be expensive.

For a randomly selected car of price \( x \), we have that \( P(20000 - q < x < 20000 + q) = 0.6 \).

b) Find the value of \( q \). Round your answer to two decimal places.

[Maximum mark: 9]

In a small village, 10 people have installed alarm systems from the same company for their homes. This alarm system will work as expected 98% of the time and ring when something is detected.

A series of burglaries happen in the neighborhood. What is the probability that:

a) All 10 alarms will work properly?

b) At least 8 of the alarms will work properly?

A squirrel lives in this village. When he runs through a garden, he will trigger the alarm system of the house 30% of the time. Today, he runs in front of all ten houses with an alarm. What is the probability that:

c) The alarm rings when the squirrel runs in front of one house.

d) The squirrel will make at least 3 alarm systems ring.

[Maximum mark: 14]

A burger stand makes two types of burgers: cheeseburgers and hamburgers. The weight \( H \) of hamburgers is normally sitrbitued with a mean of 220g and a standard deviation of 25g.

a) Find the probability that a randomly selected hamburger weighs exactly 210g.

b) Find the probability that a randomly selected hamburger weighs more than 240g.

c) 10 hamburgers were selected at random, find the probability that less than 4 of them weigh more than 240g.

The weight \( C \) of cheeseburgers is normally sitrbitued with a mean of 230g and a standard deviation of 27g.

The burger stand produces 55% of cheeseburgers.

d) Find the probability that a randomly selected burger, from all those produced at the stand, weighs more than 240g.

e) Find the probability that a randomly selected burger is a cheeseburger, given that it weighs more than 240g.