Normal Dist

    Cards (198)

    • What type of probability distribution was studied in the previous chapter?
      Discrete probability distribution
    • Why can't all possible outcomes of continuous variables be listed?
      Because continuous variables can take an infinite number of values
    • How is probability represented for continuous variables?
      By the area under a curve called the probability density function
    • What is the probability that an observation from a continuous distribution is exactly equal to a specific value?
      Zero
    • What are the two conditions for a curve to be used as a probability density function?
      • The total area under the curve must be 1.
      • The curve must not take negative values (must not go below the horizontal axis).
    • What is the main characteristic of a normal distribution?
      It has a high probability density close to the mean
    • What are some examples of variables that likely follow a normal distribution?
      Heights of adult females, lengths of leaves, widths of car doors, times taken by 12-year-old boys to run 100 m
    • What is the standard normal distribution?
      A normal distribution with mean 0 and standard deviation 1
    • What is the equation of the probability density function for the standard normal distribution?

      f(z) = 12πez22\frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}
    • What does the table of the normal distribution function provide?

      The probability that a normally distributed random variable Z is less than or equal to z
    • How do you find the probability for a positive z-value using the normal distribution table?
      Locate the row for the integer part and the column for the decimal part
    • What is the probability that an observation from a standard normal distribution is less than 1.36?
      0.91309
    • How can you estimate the probability for a z-value given to more than two decimal places?
      By using interpolation between the closest values in the table
    • How do you find the probability of a value greater than a specific z-value?
      Use the fact that the total area under the curve is 1 and subtract the probability of the z-value from 1
    • What is the relationship between negative z-values and positive z-values in the normal distribution?
      Negative z-values can be derived from positive z-values due to symmetry
    • How do you find the probability that z lies between two values?
      Use the areas to the left of the two z-values and subtract them
    • What is the process for standardizing a normal variable?
      1. Subtract the mean from the value of interest.
      2. Divide the result by the standard deviation.
      3. This gives the z-score.
    • How do you calculate the z-score for a value of 60 cm with a mean of 50 cm and standard deviation of 5 cm?
      z = 60505=\frac{60 - 50}{5} =2 2
    • What is the z-score for a value of 47 cm with a mean of 50 cm and standard deviation of 5 cm?
      z = 47505=\frac{47 - 50}{5} =0.6 -0.6
    • What is the first step to find the probability that a bird has a wingspan less than 17 cm?
      Calculate the z-score for 17 cm
    • If the mean wingspan is 14.1 cm and the standard deviation is 1.7 cm, what is the z-score for a wingspan of 17 cm?
      z = 1714.11.71.71\frac{17 - 14.1}{1.7} \approx 1.71
    • What is the probability of a wingspan less than 17 cm when the z-score is 1.71?
      0.956
    • What are the steps to find the probability for a normal distribution with given mean and standard deviation?
      1. Standardize the value to find the z-score.
      2. Use the z-score to find the corresponding probability from the standard normal distribution table.
      3. Interpret the probability based on the context of the problem.
    • How do you find the probability that a randomly selected customer has a chest measurement less than 103 cm with a mean of 101 cm and standard deviation of 5 cm?
      Calculate the z-score and then find the corresponding probability
    • What is the z-score for a chest measurement of 98 cm with a mean of 101 cm and standard deviation of 5 cm?
      z = 981015=\frac{98 - 101}{5} =0.6 -0.6
    • What is the probability that a customer has a chest measurement of 98 cm or more?
      0.726
    • How do you find the probability between two chest measurements, 95 cm and 100 cm?
      Calculate the z-scores for both measurements and find the area between them
    • What is the z-score for a chest measurement of 95 cm with a mean of 101 cm and standard deviation of 5 cm?
      z = 951015=\frac{95 - 101}{5} =1.2 -1.2
    • What is the z-score for a chest measurement of 100 cm with a mean of 101 cm and standard deviation of 5 cm?
      z = 1001015=\frac{100 - 101}{5} =0.2 -0.2
    • What is the probability between 90 cm and 110 cm for a chest measurement with mean 101 cm and standard deviation 5 cm?
      0.950
    • How do you ensure accuracy when finding probabilities from tables?
      Keep as many figures as possible during calculations
    • What should you do if a question asks for a probability less than a specific value?
      Use the value given without rounding
    • What are the key steps to solve problems involving normal distributions?
      1. Identify the mean and standard deviation.
      2. Standardize the value to find the z-score.
      3. Use the z-score to find the corresponding probability.
      4. Interpret the results in the context of the problem.
    • What is the mean and standard deviation of the normal distribution for the wingspans of a population of birds?
      Mean is 14.1 cm and standard deviation is 1.7 cm
    • How do you find the probability that an item chosen from a normal distribution with mean 19.6 cm and standard deviation 1.9 cm is less than 20.4 cm?

      Calculate the z-score and find the corresponding probability
    • What is the z-score for a measurement of 20.4 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
      z = 20.419.61.90.421\frac{20.4 - 19.6}{1.9} \approx 0.421
    • How do you find the probability that an item is more than 22.0 cm in a normal distribution with mean 19.6 cm and standard deviation 1.9 cm?
      Calculate the z-score and subtract the probability from 1
    • What is the z-score for a measurement of 22.0 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
      z = 22.019.61.91.263\frac{22.0 - 19.6}{1.9} \approx 1.263
    • How do you find the probability that a measurement is between 19.0 and 21.0 cm?
      Calculate the z-scores for both measurements and find the area between them
    • What is the z-score for a measurement of 19.0 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
      z = 19.019.61.90.316\frac{19.0 - 19.6}{1.9} \approx -0.316
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