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Probability Distributions

Probability Distributions. Continuous distributions. Sample size 24. Guess the mean and standard deviation. Dot plot sample size 49. Draw the population distribution you expect. Sample size 93. Sample size 476. Sample size 948. Mean 160 Median 161 Standard deviation 12s.

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Probability Distributions

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  1. Probability Distributions Continuous distributions

  2. Sample size 24

  3. Guess the mean and standard deviation

  4. Dot plot sample size 49

  5. Draw the population distribution you expect

  6. Sample size 93

  7. Sample size 476

  8. Sample size 948

  9. Mean 160Median 161 Standard deviation 12s

  10. Shape activity 1.10

  11. Mean = (12 + 8 + 7 + 14 + 4) ÷ 5 = 9 25 4 -5 4 7 -2 25 14 5 1 8 Calculator function -1 9 12 3 1st slide Calculate the mean Standard Deviation Given a Data Set 12, 8, 7, 14, 4 The standard deviation is a measure of the mean spread of the data from the mean. How far is each data value from the mean? (25 + 4 + 25 + 1 + 9) ÷ 5 =12.8 Square to remove the negatives Square root 12.8 = 3.58 Std Dev = 3.58 Average = Sum divided by how many values Square root to ‘undo’ the squared

  12. Looking at distributions(simulated normal distribution) • Small samples do not always have distributions like the population they come from. • When looking at distributions, a sample of 30 is much too small to give a good picture of the whole population distribution.

  13. Looking at distributions(simulated normal distribution) • Large samples do have distributions like the population they come from. • When looking at distributions, a sample of about 200 is sufficient to give a picture of the whole population distribution.

  14. Estimating mean and standard deviation To estimate mean and standard deviation, you need to know that: • The mean is pulled towards extreme values • The SD is stretched by extreme values If the distribution is approximately normal, the mean is the middle, and the SD is roughly 1/6th the range (97.8% within μ ± 3σ).

  15. Estimating mean and standard deviation for any distribution Estimating the mean: • Estimate the median and adjust towards extreme values. Estimating the standard deviation: • Estimate the median distance from the mean and adjust it (stretch it if there are extreme values).

  16. Estimate the mean and standard deviation of the age of students completing the census@school survey. Mean = 12.3 years SD = 1.8 years

  17. Words remembered in Kim’s Game Mean = 13.1 SD = 2.4 Mean = 9.0 SD = 2.8

  18. Text messages sent in a day by stage one university students Mean = 38 messages SD = 57 messages

  19. Number of pairs of shoes owned by stage one university students Mean = 10.4 pairs SD = 8.9 pairs

  20. Mean = 5.9 words SD = 2.5 words Mean = 7.0 words SD = 23 words

  21. Continuous probability graphs What are the units on the vertical axis for a continuous probability function?

  22. Continuous probability graphs are probability density functions The vertical axis measures the rate probability/x, which is called probability density. Probability density is only meaningful in terms of area.

  23. CONTINUOUS

  24. Draw the probability density function for the following

  25. bus waiting time (1) The downtown inner link bus in Auckland arrives at a stop every ten minutes, but has no set times. If I turn up at the bus stop, how long will I expect to wait for a bus? What will the distribution of wait times look like?

  26. a b c

  27. 0.1 0 10

  28. Which is more likely: a wait of between 2 and 5 minutes, or a wait of more than 6 minutes, measured to the nearest minute? 0.1 0 10

  29. Bus waiting time (2) • My own bus route (277) runs only every half hour, and isn’t as reliable as the inner link. • I know that the bus is most likely to appear on time, but could in fact turn up at any time between the time it is due and half an hour later.

  30. What is the best model for wait time, given the available information?

  31. In the real world: Uniform models are used for modelling distributions when the only information you have are maximum and minimum. Triangular models are used for modelling distributions when the only information you have are maximum, minimum and average (could be the mode).

  32. a b c

  33. What is the probability that I will have to wait longer than 20 minutes for a bus? 1 15 0 30

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