Statistics Chapter 6: Normal Distributions

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Questions and Answers

What is the mean of the standard normal distribution?

  • 0 (correct)
  • 1
  • 2
  • -1

Which tool can be used to find cumulative probabilities for normal distributions?

  • Graphing Calculator
  • Word Processor
  • Python Programming
  • Z-Table (correct)

When using the Z-Table, what does each value in the body of the table represent?

  • The cumulative area from the right
  • The probability of obtaining a z score
  • The mean of a specific z score
  • The cumulative area from the left (correct)

What is the standard deviation of the standard normal distribution?

<p>1 (A)</p> Signup and view all the answers

Which of the following tools does not allow for finding normal distribution probabilities?

<p>Text Editor (C)</p> Signup and view all the answers

How is the Z-Table organized?

<p>One page for negative z scores and one for positive z scores (A)</p> Signup and view all the answers

What command is used in Excel to find cumulative probabilities for a normal distribution?

<p>NORMDIST (C)</p> Signup and view all the answers

What is the primary purpose of the Z-Table?

<p>To provide cumulative areas for standard normal distribution (B)</p> Signup and view all the answers

What is a characteristic of a density curve?

<p>Every point on the curve must have a vertical height that is 0 or greater. (B)</p> Signup and view all the answers

Which statement accurately describes the standard normal distribution?

<p>Its graph is bell-shaped. (C)</p> Signup and view all the answers

What does the total area under a density curve represent?

<p>The total probability, which must equal 1. (C)</p> Signup and view all the answers

In the context of the standard normal distribution, what are the values of its mean and standard deviation?

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

What does a probability density function approximate?

<p>The relative frequency distribution. (D)</p> Signup and view all the answers

Which of the following options defines a continuous random variable?

<p>It represents a range of values over an interval. (A)</p> Signup and view all the answers

What does the graph of a normal distribution typically look like?

<p>It is bell-shaped. (B)</p> Signup and view all the answers

In a histogram relating to a continuous random variable, what do the heights of the bars represent?

<p>The frequency of each class. (B)</p> Signup and view all the answers

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

<p>Distance along the horizontal scale of the distribution (A)</p> Signup and view all the answers

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

<p>The area (probability) under the curve (A)</p> Signup and view all the answers

What is the probability that a randomly selected adult has a bone density test result below 1.27?

<p>0.8980 (C)</p> Signup and view all the answers

If z scores are normally distributed with a mean of 0 and a standard deviation of 1, what z score corresponds to a reading above -1.23?

<p>P(z &gt; -1.23) equals 0.8907 (C)</p> Signup and view all the answers

Which of the following statements about the standard normal distribution is true?

<p>Z scores standardize data to a common scale. (A)</p> Signup and view all the answers

What formula can be used in Excel to find the cumulative probability for a z score of 1.27?

<p>=NORM.S.DIST(1.27, TRUE) (B)</p> Signup and view all the answers

If a readability test scores follow a standard normal distribution, which of the following z scores indicates a value below average?

<p>-0.2 (A)</p> Signup and view all the answers

How can z scores be interpreted in the context of a bone density test?

<p>They show the relative position of a test result in a normal distribution. (D)</p> Signup and view all the answers

What shape does the graph of a normal distribution exhibit?

<p>Bell-shaped (D)</p> Signup and view all the answers

In a standard normal distribution, what are the values of µ and σ?

<p>µ = 0 and σ = 1 (D)</p> Signup and view all the answers

How is a uniform distribution described in terms of value distribution?

<p>Values are spread evenly. (D)</p> Signup and view all the answers

What does the total area under the density curve of a normal distribution represent?

<p>Probability, which is equal to 1 (A)</p> Signup and view all the answers

What is the probability of selecting a thermometer with a reading above -1.23 degrees?

<p>0.8907 (A)</p> Signup and view all the answers

If a voltage level has a probability of 0.25 in a uniform distribution, what does this represent?

<p>25% of values are above the voltage level (C)</p> Signup and view all the answers

What is depicted by the shaded area in the example of voltage levels?

<p>Voltage levels greater than a specific value (A)</p> Signup and view all the answers

What is the result of P(−2.00 < z < 1.50)?

<p>0.9104 (B)</p> Signup and view all the answers

Which equation represents the probability density function (PDF) of a normal distribution?

<p>$ f(x) = rac{1}{σ imes ext{sqrt}(2 ext{π})} e^{- rac{1}{2}( rac{x-µ}{σ})^2} $ (D)</p> Signup and view all the answers

What does P(z < a) represent in standard normal distribution notation?

<p>The probability that the z score is less than a. (A)</p> Signup and view all the answers

Which of the following is NOT a feature of a normal distribution?

<p>Has a finite range of values (C)</p> Signup and view all the answers

If P(z < 1.56) = 0.9406, what is P(z > 1.56)?

<p>0.0594 (C)</p> Signup and view all the answers

What is the probability of obtaining a z score less than -1.56?

<p>0.0594 (C)</p> Signup and view all the answers

Which of the following is the correct formula to calculate P(−0.85 < z < 1.56)?

<p>P(z &lt; 1.56) − P(z &lt; −0.85) (D)</p> Signup and view all the answers

What is the probability of a z score being either less than -1.65 or greater than 1.65?

<p>0.1056 (C)</p> Signup and view all the answers

For P(z > −0.85), what is the resulting probability?

<p>0.8023 (D)</p> Signup and view all the answers

Flashcards

Continuous Random Variable

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

Density Curve

A density curve is the graph of a continuous probability distribution. It's a smooth, unbroken curve that describes the probability of a variable taking on any given value.

Area under Density Curve

The total area under the density curve must equal 1. This represents the fact that the probability of the variable taking on any value within its range is 1.

Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution with a mean of 0 and a standard deviation of 1.

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Bell-shaped Curve

The graph of a standard normal distribution is bell-shaped, symmetrical around its mean of 0.

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Mean of Standard Normal

The mean of a standard normal distribution is always 0, meaning the average value of the data is centered at 0.

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Standard Deviation of Standard Normal

The standard deviation of a standard normal distribution is always 1, indicating the spread of the data points around the mean.

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Applications of Standard Normal

The standard normal distribution is used extensively in statistics because many real-world phenomena are approximately normally distributed.

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Uniform Distribution

A continuous distribution where the probability of each value is equal.

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Uniform Distribution Graph

The graph of a uniform distribution looks like a rectangle.

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Normal Distribution

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

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Area and Probability Relationship

The area under the curve of a probability distribution equals 1, representing the total probability.

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Probability and Shaded Area

The shaded area under a probability distribution graph represents the probability of a given event.

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Area-Probability Correspondence

The correspondence between area and probability allows us to calculate probabilities by calculating the area under the density curve.

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Z-Table

A table that lists the area under the standard normal curve to the left of a given z-score.

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z-score

The distance along the horizontal axis of the standard normal distribution. It represents the number of standard deviations a value is away from the mean.

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Area under the standard normal curve

The area under the curve of a standard normal distribution. It represents the probability of a value being less than or equal to a given z-score.

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P(z < value)

The probability of a z-score being less than a given value.

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P(z > value)

The probability of a z-score being greater than a given value.

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P(value1 < z < value2)

The probability of a z-score being between two given values.

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Bone Mineral Density Test

A statistical test used to identify osteoporosis, a condition causing bone fragility.

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Standardization

A method used to convert a value from a normal distribution with any mean and standard deviation to a standard normal distribution with a mean of 0 and a standard deviation of 1 by subtracting the mean and dividing by the standard deviation.

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Finding probabilities using the standard normal distribution

The process of finding probabilities using the standard normal distribution.

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Cumulative area

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

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Using the Z-Table

To find the probability associated with a specific z-score, locate the z-score in the Z-table and read the corresponding area. This area is the probability of observing a z-score less than or equal to the given value.

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P(a < z < b)

The probability that a z-score falls between two specified values.

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P(z > a)

The probability that a z-score is greater than a given value.

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P(z < a)

The probability that a z-score is less than a given value.

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P(z > -1.23)

The probability that a z-score is greater than -1.23.

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P(-2.00 < z < 1.50)

The probability that a z-score falls between -2.00 and 1.50.

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P(z < -2.00)

The probability that a z-score is less than -2.00.

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P(z < 1.50)

The probability that a z-score is less than 1.50.

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What is the probability of selecting a thermometer with a reading above -1.23 degrees?

The probability of randomly selecting a thermometer with a reading above -1.23 degrees.

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Study Notes

Chapter 6: Normal Probability Distributions

  • This chapter covers normal probability distributions.
  • It includes information on continuous random variables, density curves, the standard normal distribution, and applications of normal distributions.

Continuous Random Variable

  • A continuous random variable, represented by x, can taken on any value within a given range.
  • In this context, the heights of 5,000 female students are used as an example.
  • Table 1 presents frequency and relative frequency distributions for student heights (in inches) across different ranges.

Density Curve

  • A density curve visually represents a continuous probability distribution.
  • Key properties: the total area under the curve equals 1; every point on the curve has a vertical height of 0 or greater (the curve cannot go below the x-axis).
  • (Figure 1) illustrates a histogram and polygon for the relative frequency distribution in Table 1. This is an approximation of the continuous probability distribution curve for the variable x.

The Standard Normal Distribution

  • The standard normal distribution, denoted by Z, is a specific type of normal distribution.
  • It has a mean of 0 and a standard deviation of 1, i.e. Z ~ N(0,1)
  • Its graph is bell-shaped and symmetrical.

Normal Distribution

  • A continuous random variable is normally distributed if its distribution is bell-shaped and symmetrical following the equation: f(x) = e^(-(x-μ)^2/(2σ^2)) / (σ√2Ï€)
  • A uniform distribution is a continuous random variable whose values are evenly spread within the range of probabilities.

Using Area to Find Probability

  • The total area beneath the density curve is always equal to 1.
  • This means there's a direct relationship between the area under the curve and probability.
  • Example illustration: Finding the probability of a randomly chosen voltage level being above 124.5 volts, given a specific uniform distribution

Finding Probabilities when given z Scores

  • Z-Table is used to calculate probabilities in normal distributions. other methods include using formulas and tools like STATDISK, Minitab, Excel, and TI/83/84 Plus.

Using Z-Table

  • Z-Table is a table used to find cumulative areas under the standard normal curve. It's used to determine probabilities for different regions.
  • The table provides probabilities for different z-score values. Different software packages provide the calculated values, including Excel, Minitab, and STATDISK.

Finding a z Score when given a Probability

  • The process involves drawing a bell-shaped curve.
  • Using the cumulative area, locating the closest probability in the Z-Table to find the corresponding z-score.

Applications of Normal Distributions

  • The chapter introduces converting non-standard normal distributions to standard normal distributions using the formula Z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation.
  • This allows using the Z-table for probability calculations involving non-standard normal distributions.
  • Examples involve calculating probabilities and finding specific values (e.g., weights, heights) given data characteristics.

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Chapter 6: Normal Probability Distributions PDF

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