Probability and Expected Values Quiz

Questions and Answers

A student guesses the answer to every question. Let X be the number of correct answers. They will expect to get ______ correct.

The probability of winning a prize in a game of chance is ______.

0.35

In a sample of n children, 70% said yes to using social media. The sample proportion p ̂ is ______.

0.7

The heights of the class of 2024 are normally distributed with an average of ______ cm.

175 Signup and view all the answers

60% of students catch public transportation to school. That day, the probability that at least one of the selected students caught public transport is ______.

greater than 0 Signup and view all the answers

The probability that a full forward will kick a goal is ______.

0.15 Signup and view all the answers

If there are 74 students in the cohort, we would expect ______ of them to be shorter than 168cm.

a number Signup and view all the answers

The standard deviation of the heights in the class is ______ cm.

6 Signup and view all the answers

The probability function f(x) for the continuous random variable X is given as f(x) = {█(k______(4-x),0≤x≤4@@0, otherwise).

x Signup and view all the answers

In a particular city, it was surveyed that ______% of people have a driver’s license.

60 Signup and view all the answers

15% of coke cans have less than ______ mL of coke.

372 Signup and view all the answers

The mean and standard deviation of the amount of coke in a coke can are found using the ______ distribution.

normal Signup and view all the answers

A survey showed that 237 from a random sample of ______ answered yes to a certain question.

500 Signup and view all the answers

The confidence intervals for the proportion of undergraduates are calculated at ______%, 95%, and 99%.

90 Signup and view all the answers

Study Notes

Question 1: Probability and Expected Values

Exam consists of 30 multiple-choice questions with 3 possible answers each.
If a student guesses, the expected number of correct answers, E(X) or μ, is calculated as (μ = \frac{1}{3} \times 30 = 10).
Variance, Var(X), can be computed using the formula for a binomial distribution: (Var(X) = n \cdot p \cdot (1 - p) = 30 \cdot \frac{1}{3} \cdot \frac{2}{3} = 20).
Standard deviation is the square root of variance: (SD(X) = \sqrt{Var(X)} = \sqrt{20} \approx 4.47).

Question 2: Probability in Games of Chance

Winning probability in a game is 0.35.
Aim is to find the minimum number of games to play for winning at least twice with a probability greater than 0.9.

Question 3: Sample Proportion and Margin of Error

A survey indicates 70% of primary school children use social media.
Sample proportion (p̂ = 0.70).
Margin of error (E) at 95% confidence level is expressed as (E = z \cdot \sqrt{\frac{p̂(1 - p̂)}{n}}), where (z) corresponds to the z-value for 95% confidence.
Tripling the sample size (n) reduces the margin of error (E) by about 29%.

Question 4: Normal Distribution of Heights

Heights in a class are normally distributed with mean 175 cm and standard deviation 6 cm.
To find the expected number of students shorter than 168 cm, utilize the z-score formula and the normal distribution table.

Question 5: Probability with Public Transport

60% of students use public transport; random selection of 4 students.
Probability that at least one student caught public transport can be found using (P(X ≥ 1) = 1 - P(X = 0)).

Question 6: Probability of Kicking Goals in AFL

Probability of a goal for a full forward is 0.15 with 10 kicks.
Compute probabilities for:
- Kicking a goal every time: (P(X = 10) = (0.15)^{10}).
- Kicking at least one goal: Use complementary probability.
- Kicking more than one goal given at least one goal: Conditional probability (P(X > 1 | X ≥ 1)).

Question 7: Continuous Random Variable

(f(x) = kx(4 - x)) for (0 ≤ x ≤ 4), find value of constant (k).
Calculate expected value (E(X)) using the probability function.

Question 8: Driver's License Probability

60% of surveyed individuals have a driver's license.
Approximate probability for more than 65% of a sample of 200 people using normal approximation to the binomial distribution.

Question 9: Normal Distribution of Coke Cans

15% of cans < 372 mL, and 10% > 379 mL.
Find mean and standard deviation of coke amount in a can.
To determine the proportion of cans with less than 377 mL, use the derived normal distribution parameters.

Question 10: Confidence Intervals for Postgraduate Intent

Survey of 500 individuals shows 237 want to pursue postgraduate studies.
Calculate confidence intervals for proportions at 90%, 95%, and 99% levels, comparing the intervals statistically.

Description

Test your understanding of probability concepts and expected values with this engaging quiz. Covering topics from multiple-choice guessing to games of chance and sample proportions, this quiz will challenge your grasp on statistical measures. Perfect for students looking to enhance their skills in probability theory.