Math 1040 Chapter 7

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bpulsipher
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298528
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Math 1040 Chapter 7
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2015-03-19 11:17:56
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A list of keywords and principles from Chapter 7.
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  1. Define uniform probability distribution.
    Any two intervals of equal length are equally likely, we say that the random variable "X" follows a uniform probability distribution.
  2. What is a probability density function (pdf)? (Hint: In chapter 7, 'pdf' does not only stand for "probability distribution function" as it did in chapter 6)
    An equation used to compute probabilities of continuous random variables. It must satisfy the following two properties:

    (1) the total area under the graph of the equation over all possible values of the random variable must equal 1.

    (2) the heights of the graph of the equation must be greater than or equal to "0" for all possible values of the random variable. That is, the graph of the equation must lie on or above the horizontal axis for all possible values of the random variable.
  3. If the possible values of a uniform density function go from 0 through n, what is the height of the rectangle?
    Area of a rectangle is the product of height and width. Therefore the height must be 1/n.
  4. What does the area under the graph of a probability density function over an interval represent?
    the probability of observing a value of the random variable in that interval.
  5. T or F

    Not all continuous random variables follow a uniform distribution.
    True
  6. what value of x is associated with the peak of W normal curve (continuous random variables)?
    the mode

    for symmetric distributions with a single peak, such as the normal distribution, the mean = median = mode. Because of this, the mean, , is the high point of the graph of the distribution.
  7. What values of x are associated with the inflection points of a normal curve (continuous random variables)?
     and  because these are the points on the distribution curve where the curvature changes.
  8. What happens to the graph (continuous random variables in a data set) when the mean is shifted to the right? Left? What happens to the same graph when the standard deviation increases? Decreases?
    when the mean is shifted to the right, the normal distribution retains the same shape and shifts to the right and to the left when the mean shifts to the left.

    when the standard deviation increases, the shape of the normal distribution flattens out and becomes wider. When the standard deviation decreases, the normal distribution compresses and becomes steeper.
  9. State the seven properties of the normal curve.
    1) The normal curve is symmetric about its mean, 

    2) Because mean = median = mode (for qualitatitve data), the normal curve has a single peak and the highest point occurs at 

    3) The normal curve has inflection points at  and 

    4) The area under the normal curve is 1.

    5) The area under the normal curve to the right of  equals the area under the normal curve to the left of , which equals 0.5.

    6) As "x" increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As "x" decreases without bound (gets more and more negative), the graph approaches, but never reaches, the horizontal axis.

    • 7) The Empirical Rule:
    • Approximately 68% of the area under the normal distribution curve is between  and , approximately 95% of the area is between  and , and approximately 99.7% of the area is between  and .
  10. Suppose that a random variable X is normally distributed with mean  and standard deviation . Give two representations for the area under the normal curve for any interval of values of the random variable X.
    1) The proportion of the population with the characteristic described by the interval of values,

    or

    2) The probability that a randomly selected individual from the population will have the characteristic described by the interval of values.
  11. 7.2 (small data sets)

    A normal probability plot is...
    ...a graph that plots observed data versus normal scores.
  12. 7.2 (small data sets)

    A normal score is...
    ...the expected z-score of the data value, assuming that the distribution of the random variable is normal. The expected z-score of an observed value depends on the number of observations in the data set.
  13. 7.2 (small data sets)

    What are the four steps to follow when drawing a Normal Probability Plot by hand? What is the difference between a z-score and a fi-score?
    Step 1 - Arrange the data in ascending order.

    Step 2 - Compute fi=i−0.375n+0.25 where i is the index (the position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is fi.

    Step 3 - Find the z-score corresponding to fi from Table V.

    Step 4 - Plot the observed values on the horizontal axis and the corresponding expected z-scores on the vertical axis.

    The value of fi represents the expected area to the left of the ith observation when the data come from a population that is normally distributed. This is just like the idea of finding the area to the left of an expected z-score under a normal distribution (observations vs prediction based on the average).

    Once we determine each fi, we find the z-scores corresponding to f1 (represents z1 value for observed values), f2 (represents z2 for observed values), and so on.
  14. 7.2 (small data sets)

    Values of normal random variables and their z-scores are linearly related (x=μ+zσ), so a plot of observations of normal variables against their expected z-scores will be linear. We conclude the following:

    T or F

    If sample data are taken from a population that is normally distributed, a normal probability plot of the observed values versus the expected z-scores will be approximately linear.
    True
  15. 7.2 (small data sets)

    T or F

    If the linearly correlation coefficient between the observed values and expected z-scores is greater than the critical value found in Table VI, then it is reasonable to conclude that the data could come from a population that is normally distributed.
    True

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