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. (A)</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. (D)</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. (C)</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. (A)</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% (C)</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 (D)</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 (A)</p> Signup and view all the answers

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

<p>Relative frequencies (B)</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 (C)</p> Signup and view all the answers

What is the purpose of a density curve in statistics?

<p>To represent a continuous probability distribution (A)</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 (D)</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 (C)</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 (A)</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 (C)</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) (A)</p> Signup and view all the answers

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

<p>0.0228 (A)</p> Signup and view all the answers

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

<p>0.7364 (D)</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) (B)</p> Signup and view all the answers

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

<p>0.0670 (C)</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 (B)</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 (A)</p> Signup and view all the answers

What shape does the graph of a normal distribution have?

<p>Symmetric and bell-shaped (A)</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 (B)</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 (C)</p> Signup and view all the answers

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

<p>STATDISK (A)</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 (B)</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 (D)</p> Signup and view all the answers

How many pages is the Z-Table composed of?

<p>Two pages for negative and positive scores (A)</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 (A)</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 (A)</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 (C)</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 (A)</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 (B)</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 (C)</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 (C)</p> Signup and view all the answers

Flashcards

Continuous Random Variable

A variable that can take on any value within a given range.

Density Curve

A graph that represents a probability distribution for a continuous random variable. It shows the relative likelihood of different values.

Standard Normal Distribution

A symmetrical, bell-shaped distribution that has a mean of 0 and a standard deviation of 1.

Mean (µ)

The average value of a distribution, representing the central tendency of the data.

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Standard Deviation (σ)

A measure of the spread or variability of a distribution, indicating how far data points typically are from the mean.

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Total Area Under Density Curve

The total area under the density curve of a probability distribution must always equal 1.

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Vertical Height of Density Curve

The vertical height of a density curve at any point represents the probability density at that value.

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Probability Density Function

The graph of a probability distribution for a continuous random variable.

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Normal Distribution

A continuous distribution with a bell-shaped, symmetric graph. It's described by a specific mathematical formula.

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Uniform Distribution

A continuous distribution where all values have an equal chance of occurring within a defined range. Its graph resembles a rectangle.

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Area and Probability

The area under a probability density curve corresponds to the probability of a random variable falling within a specific range.

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Area and Probability (Standard Normal)

The area under the standard normal distribution curve represents the probability of a random variable falling within a specified interval.

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Transforming Normal Distributions

The standard normal distribution is often used to transform other normal distributions into a standard form for easier comparisons and calculations.

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Standardization

The process of converting a normal distribution to a standard normal distribution by subtracting the mean and dividing by the standard deviation.

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Z-score

A standardized score that represents the number of standard deviations a data point is away from the mean in a standard normal distribution.

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Z-Table

A table that provides the area under the standard normal curve for different z-scores.

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Cumulative Area (Z-Table)

The area under the standard normal curve to the left of a specific z-score. It represents the probability of getting a value less than or equal to that z-score.

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Finding Probabilities with Z-Scores

The process of finding the area under the standard normal curve using a Z-Table or other statistical software.

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STATDISK

A statistical software package that can be used to find probabilities in a normal distribution, including using Z-Scores.

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Minitab

A statistical software package that can be used to find probabilities in a normal distribution, including using Z-Scores.

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Excel

A spreadsheet application that can be used to find probabilities in a normal distribution, including using Z-Scores.

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P(z > a)

The probability that a randomly selected value from a standard normal distribution is greater than a specific value.

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P(z < a)

The probability that a randomly selected value from a standard normal distribution is less than a specific value.

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P(a < z < b)

The probability that a randomly selected value from a standard normal distribution falls between two specific values.

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P(-2.00 < z < 1.50)

The probability that a randomly selected value from a standard normal distribution is between -2.00 and 1.50 degrees.

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P(z > -1.23)

The probability that a randomly selected thermometer has a reading above -1.23 degrees.

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P(z < 1.56)

The probability that a randomly selected thermometer has a reading below 1.56 degrees.

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What is the z-score?

In a standard normal distribution, the z-score represents the distance of a specific data point from the mean, measured in units of standard deviations.

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What does the area under the standard normal curve represent?

The area under the curve of a standard normal distribution represents the probability of observing values less than or equal to the corresponding z-score.

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What is the purpose of the Z-Table?

The Z-Table provides a lookup table for finding probabilities associated with different z-scores. It shows the area under the curve to the left of a given z-score.

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How do you find the hundredths place of a z-score on the Z-Table?

The part of the z-score denoting hundredths is found across the top row of the Z-Table.

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What does P(z < 1.27) represent?

The probability that a randomly selected person has a bone density test result below 1.27 is represented by P(z < 1.27) and equals 0.8980.

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How can z-scores be used to find probabilities for specific ranges?

The z-score can be used to determine the probability of an event occurring within a specific range of values. For example, P(z > -1.23) represents the probability of a thermometer reading above -1.23 degrees.

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Explain the mean and standard deviation of a standard normal distribution.

In the context of a standard normal distribution, the mean is 0, and the standard deviation is 1. This means that the distribution is centered at 0, and the spread of the data is measured by 1 unit.

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What's the difference between z-scores and areas in the standard normal distribution?

When working with a standard normal distribution, always differentiate between z-scores and areas. The z-score represents the distance from the mean, while the area corresponds to the probability under the curve.

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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√2Ï€'.

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

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