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# MDM4U Grade 12 Data Management: Probability Distributions

Created by
@SteadfastCyclops

## Questions and Answers

### What is the range of movie lengths in the scenario described in question 1?

80 minutes to 125 minutes

### What is the probability that a randomly selected movie will run for less than 95 minutes?

$rac{15}{45} = rac{1}{3}$

X ~ N(25, 2.5)

### What is the condition for approximating a binomial distribution with a normal distribution?

<p>n × p ≥ 5 and n × (1 - p) ≥ 5</p> Signup and view all the answers

### Determine the probability that a student randomly selected from the class will have a test score between 69% and 81%.

<p>Approximately 0.6827 (using the standard normal distribution table)</p> Signup and view all the answers

### What is the probability that a student randomly selected from the class will have a test score greater than 88%?

<p>Approximately 0.1587 (using the standard normal distribution table)</p> Signup and view all the answers

### What is the shape of the graph for a continuous uniform distribution?

<p>A rectangle</p> Signup and view all the answers

### What is the mean of a normal distribution described as X ~ N(25, 2.5)?

<p>25</p> Signup and view all the answers

## Study Notes

### Continuous Probability Distributions

#### Uniform Distribution

• A uniform distribution is a continuous probability distribution where every possible value has an equal chance of being selected.
• Example: movie lengths ranging evenly between 80 minutes and 125 minutes.
• Graph: rectangular shape with equal heights and widths.

#### Normal Distribution

• A normal distribution is a continuous probability distribution that is symmetric and bell-shaped.
• Parameters: mean (μ) and standard deviation (σ).
• Example: test scores with a mean of 25 and a standard deviation of 2.5.
• Notation: X ~ N(μ, σ)
• Graph: bell-shaped curve with mean at the center and standard deviation marking the spread.

#### Approximating Binomial Distributions with Normal Distributions

• A binomial distribution can be approximated using a normal distribution if:
• n (number of trials) is large (> 30).
• p (probability of success) is close to 0.5.
• Examples:
• n = 50, p = 0.92: can be approximated using normal distribution.
• n = 75, p = 0.2: can be approximated using normal distribution.
• n = 25, p = 0.4: cannot be approximated using normal distribution.

#### Calculating Probabilities with Normal Distributions

• To calculate the probability of an event, find the z-score and use a standard normal distribution table or calculator.
• Example: test scores with a mean of 75% and a standard deviation of 6%.
• Probability of a score between 69% and 81%: find z-scores and calculate probability.
• Probability of a score greater than 88%: find z-score and calculate probability.
• Graph: sketch the normal distribution curve and shade the desired area.

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## Description

A formative quiz on continuous probability distributions for Grade 12 Data Management students, covering topics such as graph sketching and probability calculation.

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