MDM4U Grade 12 Data Management: Probability Distributions
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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?

<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

Flashcards

Movie Length Range

Movie lengths range from 80 to 125 minutes.

Probability < 95 min

15 out of 45 movies are shorter than 95 minutes = 1/3 probability.

Normal Dist. Notation

X ~ N(25, 2.5) describes a normal distribution with mean 25 and standard deviation 2.5.

Binomial Normal Approx

Approximating a binomial with a normal distribution requires np >= 5 and n(1-p) >= 5.

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Test Score (69-81%)

Probability is ~0.6827 for scores between 69% and 81%.

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Test Score > 88%

Probability is ~0.1587 for scores above 88%.

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Uniform Dist. Shape

A continuous uniform distribution has a rectangular graph shape.

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Normal Mean

The mean of a normal distribution X ~ N(25, 2.5) is 25.

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Movie Lengths

A range of movie lengths between 80 and 125 minutes.

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Probability Calculation

Calculating probabilities involving fractions of movies, or event occurrence in a set.

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Normal Distribution

A bell-shaped probability distribution.

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Binomial Distribution

Probability distribution of a binomial experiment.

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Standard Normal Distribution

A normal distribution with mean 0 and standard deviation 1.

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Continuous Uniform Distribution

Probability distribution where all values in a given range are equally likely.

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Mean

The average value.

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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.

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