## 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?

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

What is the notation used to describe a normal distribution with a mean of 25 and a standard deviation of 2.5?

X ~ N(25, 2.5)

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

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

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

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

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

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## 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.