We assume in this part that the student uses the bicycle to go to his school. When he uses the bicycle, his travel time, expressed in minutes, between his home and his school is modeled by a random variable $T$ which follows a normal distribution with mean $\mu = 17$ and standard deviation $\sigma = 1.2$.

1. Determine the probability that the student takes between 15 and 20 minutes to get to his school.
2. He leaves his home by bicycle at 7:40 a.m. What is the probability that he is late for school?
3. The student leaves by bicycle. Before what time must he leave to arrive on time at school with a probability of 0.9? Round the result to the nearest minute.

According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
The store manager wants to know the number $n$ of budget padlocks he must have in stock at the beginning of the month, so that the probability of running out of stock during the month is less than 0.05. The stock is not replenished during the month.
Determine the smallest integer value of $n$ satisfying this condition.

Part A
As part of its activity, a company regularly receives quotation requests. The amounts of these quotations are calculated by its secretariat. A statistical study over the past year leads to modelling the amount of quotations by a random variable $X$ which follows the normal distribution with mean $\mu = 2900$ euros and standard deviation $\sigma = 1250$ euros.

1. If a quotation request received by the company is chosen at random, what is the probability that the quotation amount exceeds 4000 euros?
2. In order to improve the profitability of its activity, the entrepreneur decides not to follow up on $10\%$ of requests. He discards those with the lowest quotation amounts. What must be the minimum amount of a requested quotation for it to be taken into account? Give this amount to the nearest euro.

At the beginning of 2021 in Germany, approximately 320000 cars with purely electric drive and 280000 plug-in hybrids were registered, thus a total of approximately 600000 cars with electric motors. The share of cars with electric motors in the total stock of all cars registered in Germany was around $1.2 \%$. Determine the number of cars that must be randomly selected from this total stock so that with a probability of more than $97 \%$ at least one car with purely electric drive is among them.

The length of the Pacific sardine (Sardinops sagax) can be considered a random variable with normal distribution with mean 175 mm and standard deviation 25.75 mm.\ a) (1 point) A canning company for this variety of sardines only accepts as quality sardines those with a length greater than 16 cm. What percentage of the sardines caught by a fishing vessel will be of the quality expected by the canning company?\ b) ( 0.5 points) Find a length t $< 175 \mathrm {~mm}$ such that between t and 175 mm there are 18\% of the sardines caught.\ c) (1 point) At sea, sardines are processed in batches of 10. Subsequently, sardines from each batch that are smaller than 15 cm are returned to the sea as they are considered small. What is the probability that in a batch there is at least one sardine returned as small?