Normal Probability Distributions
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Questions and Answers

What does a z score represent in the context of the standard normal distribution?

  • The raw score before any transformations.
  • The total area under the curve for the normal distribution.
  • A measure of how far a data point is from the mean, expressed in standard deviations. (correct)
  • An index number used for ranking test results.

    • What is the mean and standard deviation of the population of z scores in the context of the bone density test?

  • 0 and 1 respectively. (correct)
  • 0 and 0 respectively.
  • 1 and 0 respectively.
  • 1 and 1 respectively.

    • What does the probability P(z < 1.27) = 0.8980 indicate about bone density results?

  • 89.80% of people have bone density levels below 1.27. (correct)
  • Only 10.20% of people have a lower bone density than 1.27.
  • There is an equal chance of having a density result above or below 1.27.
  • 71.20% of the population has a bone density above 1.27.

    • How can one determine the probability of a z score being greater than a specific value?

    <p>It requires finding the area to the left of that z score and subtracting from 1.</p> Signup and view all the answers

    What does the top row of the Z-Table denote?

    <p>The part of the z score denoting hundredths.</p> Signup and view all the answers

    If the average reading of thermometers is 0 degrees with a standard deviation of 1 degree, what is P(z > -1.23)?

    <p>Approximately 0.8907.</p> Signup and view all the answers

    In interpreting z scores, what must be differentiated from area in the context of the Z-Table?

    <p>The z score represents distance along the horizontal scale.</p> Signup and view all the answers

    If the probability of selecting a randomly chosen adult with a bone density test result less than 1.27 is 0.8980, what percentage corresponds to this probability?

    <p>89.80%</p> Signup and view all the answers

    What is the total area under a density curve for a continuous probability distribution?

    <p>Equal to 1</p> Signup and view all the answers

    Which of the following is NOT a property of the standard normal distribution?

    <p>Standard deviation is greater than 1</p> Signup and view all the answers

    What can be used as probabilities in a relative frequency distribution?

    <p>Relative frequencies</p> Signup and view all the answers

    Which statement about the vertical heights on a density curve is true?

    <p>They must be 0 or greater</p> Signup and view all the answers

    What is the purpose of a density curve in statistics?

    <p>To represent a continuous probability distribution</p> Signup and view all the answers

    Regarding the smoothed polygon displayed in Figure 1, it is best described as what?

    <p>An approximation of the probability distribution curve</p> Signup and view all the answers

    Which of the following statements about continuous random variables is correct?

    <p>They can take any value within a given interval</p> Signup and view all the answers

    What does a bell-shaped graph in the standard normal distribution indicate?

    <p>Symmetrical distribution around the mean</p> Signup and view all the answers

    What is the probability that a randomly selected thermometer has a reading above -1.23 degrees?

    <p>0.8907</p> Signup and view all the answers

    To find the probability that a thermometer's reading is between -2.00 and 1.50 degrees, which calculation is correct?

    <p>P(z &lt; 1.50) - P(z &lt; -2.00)</p> Signup and view all the answers

    What is the probability P(z < -2.00)?

    <p>0.0228</p> Signup and view all the answers

    Calculate P(-0.85 < z < 1.56). What value represents this probability?

    <p>0.7364</p> Signup and view all the answers

    Which notation correctly expresses the probability that the z-score is greater than a given value a?

    <p>P(z &gt; a)</p> Signup and view all the answers

    For what probability is P(z < -1.65 or z > 1.65) calculated?

    <p>0.0670</p> Signup and view all the answers

    What does the Z-Table provide for a given z score?

    <p>The probability of obtaining a z score lower than the given value</p> Signup and view all the answers

    What is the mean and standard deviation in a standard normal distribution?

    <p>Mean = 0, Standard Deviation = 1</p> Signup and view all the answers

    What shape does the graph of a normal distribution have?

    <p>Symmetric and bell-shaped</p> Signup and view all the answers

    What is the relationship between area and probability in a uniform distribution?

    <p>The total area under the curve equals 1 and corresponds to probabilities</p> Signup and view all the answers

    When using Excel to find probabilities in a normal distribution, which function should be used?

    <p>NORMAL.DIST</p> Signup and view all the answers

    Which tool can be used to find normal distribution areas apart from the Z-Table?

    <p>STATDISK</p> Signup and view all the answers

    If a random variable has a standard normal distribution, what are its parameters?

    <p>µ = 0, σ = 1</p> Signup and view all the answers

    What type of distribution has values that are spread evenly over the range of probabilities?

    <p>Uniform Distribution</p> Signup and view all the answers

    How many pages is the Z-Table composed of?

    <p>Two pages for negative and positive scores</p> Signup and view all the answers

    What does f(x) = (1 / √(2πσ^2)) * e^(-(x-µ)^2 / (2σ^2)) represent?

    <p>The probability density function for a normal distribution</p> Signup and view all the answers

    Which of the following is true about the cumulative areas in the Z-Table?

    <p>They represent the area from the left up to the specific z score</p> Signup and view all the answers

    Which method would you NOT use for calculating probabilities related to a normal distribution?

    <p>Employing linear regression</p> Signup and view all the answers

    Which of the following statements about the area under the density curve is true?

    <p>The total area under the curve is equal to 1 for all distributions</p> Signup and view all the answers

    What does each value in the body of the Z-Table represent?

    <p>Cumulative area to the left of the z score</p> Signup and view all the answers

    In the example given about finding the probability of voltage levels greater than 124.5 volts, what does the shaded area represent?

    <p>The probability exceeding 124.5 volts</p> Signup and view all the answers

    What characteristic distinguishes a uniform distribution from a normal distribution?

    <p>A uniform distribution has equal probabilities while normal does not</p> Signup and view all the answers

    Study Notes

    Normal Probability Distributions

    • Normal distributions are bell-shaped and symmetrical probability distributions.
    • A continuous random variable follows a normal distribution if it can be described using a specific equation.
    • The equation for a normal distribution is complex and often represented by the symbol f(x).
      • It utilizes components like 'e' (Euler's number), 'σ', and 'σ√2π'.

    Continuous Random Variable

    • A continuous random variable, represented by 'x', can take on any value within a given range.
    • In this context, 'x' represents the heights of female students at a university.
    • A table (Table 1) shows frequency and relative frequency of student heights (in inches).
    • Heights span from 60 to 71 inches.
    • The relative frequencies in Table 1 can be treated as probabilities.

    Density Curve

    • A density curve visually displays a continuous probability distribution.
    • The total area under a density curve always equals 1.0.
    • Every point on the curve has a vertical height that is greater than or equal to zero.
    • Histograms and polygons, as shown in Figure 1, approximate density curves.

    The Standard Normal Distribution

    • The standard normal distribution has a unique bell shape.
    • Its mean is zero (μ = 0).
    • Its standard deviation is one (σ = 1).
    • This standardized distribution is often represented using the variable 'Z'.
      • Z is often used for calculations and comparisons in statistical problems.

    Normal Distribution

    • A uniform distribution occurs when values are evenly spread throughout a range of probabilities, creating a rectangular graph.
    • The total area under the density curve of any probability distribution always equals 1, creating a direct relationship between area and probability.

    Using Z-Table

    • A Z-table is a helpful resource for finding probabilities/areas associated with specific Z-scores (that is, when calculating the probability/area within a specific region below/above/between specific z-scores).
    • The table itself shows an array of z scores and associate probabilities.
    • Each value is organized according to scores and the table is used to find the cumulative area from the left of a specific z score.
      • The result will correspond to a probability.
    • This table is designed for standard normal distributions (i.e., μ = 0 and σ = 1).

    Finding a Z Score

    • Z-scores are used to show locations on a standard normal distribution.
    • To find a z score, use the cumulative area from the left to find associated z scores from the table.

    Applications of Normal Distributions

    • Normal distributions are commonly encountered in many fields.
    • There are methods for converting non-standard normal distributions to standard normal distributions.
    • The formula z = (x - μ)/σ is used for conversions.
    • This allows non-standard normal distributions to be analysed using Z-Table.

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    Chapter 6: Normal Probability Distributions PDF

    Description

    This quiz explores normal probability distributions and continuous random variables, focusing on their characteristics and mathematical representations. You will learn about the density curve and how it relates to probabilities. Prepare to analyze data on student heights and better understand statistical methods.

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