Level 2 Probability

(a)

A mathematics department surveyed 1,483 Year 12 (Level 2) students about any classes or courses they took leading up to their final exam and their final exam result (Pass or Fail).

Table 1: Study Method vs. Exam Result

(i) What proportion of all students surveyed studied using an Online Course and passed their exam?

(ii) What is the probability that a randomly chosen student who failed the exam studied with a Private Tutor?

(iii) If two students who used a Private Tutor were randomly selected from this study, what is the probability that they both failed the exam?

(iv) A university study claims that a student using an Online Course is more than 5.5 times as likely to fail the exam as a student using a Private Tutor. Determine if this claim is supported by the data, and briefly state whether this necessarily means that the Online Course is a less effective study method.

(b)

The same 1,483 Year 12 students were also asked for an estimate of their total hours studied for the exam (Less than 10 hours or 10 or more hours). Use the information below to complete the table.

Table 2: Study Method vs. Total Hours Studied

• The proportion of students who used a Private Tutor and studied Less than 10 Hrs was $\frac{1}{10}$.
• $80\%$ of students who used an Online Course studied 10 or more Hrs.
• The total number of students who studied 10 or more Hrs in the School Class group ($\mathbf{B}$) is equal to the total number of students who used an Online Course ($402$) minus the number of students who studied Less than 10 Hrs in the Private Tutor group ($\mathbf{C}$).

(i) Complete the entries A, B, C, D, E, F in the table.

(ii) If there are 12,000 students in Year 12, find the difference in the expected number of students who studied Less than 10 Hrs between those who used the Private Tutor and those who used the Online Course.

(a)

The battery life of a particular brand of smartphone is known to be normally distributed with a mean of 18.5 hours and a standard deviation of 2.4 hours.

(i) What is the probability that a randomly selected phone has a battery life of more than 21 hours?

(ii) The manufacturer claims that 90% of their phones have a battery life of at least $x$ hours. Find the value of $x$.

(iii) Given that a randomly selected phone has a battery life of more than 20 hours, what is the probability that it has a battery life of more than 22 hours?

(b)

The company collects data from a large sample of phones and records the following information about battery life. The data is known to be normally distributed.

(i) If two phones are randomly selected what is the probability that both have a battery life of less than 17.2 hours?

(ii) Explain why the mean battery life is 19.0 hours.

(iii) Use the data in the table to estimate the standard deviation of the battery life.

(c)

Sarah works for a mobile app development company and is analyzing daily screen time data from users of their productivity app. The company's marketing materials claim that the average user spends 5.2 hours per day on their smartphone, with a standard deviation of 1.4 hours. Sarah collects data from 48 randomly selected app users over one month and records their average daily screen time in the graph below.

How do Sarah's results compare to a normal distribution model with a mean of 5.2 hours and a standard deviation of 1.4 hours?

You should discuss at least TWO of centre, shape, and spread in your response.

(a)

A busy coffee shop recorded 1100 orders over one week, categorized by time of day and drink type.

(a) What proportion of all orders were for espresso-based drinks in the morning?

(b) Is a customer is more likely to order an espresso-based drink in the morning than in the afternoon? Support your answer with appropriate probability calculations.

(b)

The same coffee shop tracked customer orders and accuracy over several weeks.

The survey found that:

• 45% of customers order espresso-based drinks (lattes, cappuccinos, flat whites)
• 30% of customers order americanos
• One in four customers order tea

For coffee orders (espresso-based and americanos), the size distribution is:

• 60% of coffee orders are large
• 40% of coffee orders are small

The order accuracy rates vary by drink type and size:

• Small espresso-based drinks are wrong 15% of the time
• Large espresso-based drinks are wrong 22% of the time
• Small americanos are wrong 10% of the time
• Large americanos are wrong 14% of the time
• Tea orders are wrong 25% of the time

(i) Calculate the probability that a randomly selected customer orders a large americano and receives the correct order

(ii) Calculate the probability that a randomly selected customer orders a small espresso-based drink and receives the wrong order

(iii) Calculate the probability that a randomly selected customer receives the correct order

(iv) The coffee shop serves 1258 customers per day. The manager wants to ensure at least 1050 customers receive correct orders each day. They plan to improve the accuracy of large espresso-based drinks. What percentage of the time must large espresso-based drinks be correct to meet their goal of 1050 correct orders?