Normal Distribution and Empirical Rule
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Questions and Answers

What does a Normal distribution's theoretical density curve represent?

  • A linear function with a constant slope
  • A function that is described by a specific mathematical formula (correct)
  • A graph that has a peak at a single value
  • A curve that only includes positive values

    • Which statement about the characteristics of the Normal curve is true?

  • The curve cannot be symmetrical
  • The mean and median are always different
  • It has a single peak at the mean (correct)
  • The area under the curve is infinite

    • What percentage of data falls within 2 standard deviations of the mean in a Normal distribution?

  • 68%
  • 50%
  • 99%
  • 95% (correct)

    • What is the standard Normal distribution characterized by?

    <p>Mean of 0 and standard deviation of 1</p> Signup and view all the answers

    What does it mean for a Normal distribution to be asymptotic?

    <p>It approaches but never touches the x-axis</p> Signup and view all the answers

    Which of the following statements is NOT true about the Normal distribution?

    <p>It can fit real-life data exactly</p> Signup and view all the answers

    If a Normal distribution has a mean of 50 and a standard deviation of 5, what range contains approximately 68% of the data?

    <p>45 to 55</p> Signup and view all the answers

    What is the significance of the mean and standard deviation in a Normal distribution?

    <p>They determine the shape and spread of the distribution</p> Signup and view all the answers

    What is the median time spent playing outdoors for these children?

    <p>1.70 hrs/day</p> Signup and view all the answers

    Using the Empirical Rule, what is the calculated range for almost all children spending time outdoors?

    <p>0.35 to 3.05 hrs/day</p> Signup and view all the answers

    What does the Interquartile Range (IQR) represent in this context?

    <p>The time range for the middle 50% of children</p> Signup and view all the answers

    What is the 80th percentile of the distribution in this context?

    <p>2.05 hrs/day</p> Signup and view all the answers

    What is the value of the first quartile (Q1) for the distribution, given the z-score?

    <p>1.45 hrs/day</p> Signup and view all the answers

    How much time do the top 25% of children spend outdoors compared to the median?

    <p>More than the median</p> Signup and view all the answers

    What is the significance of using the Empirical Rule in this context?

    <p>To estimate the spread of data</p> Signup and view all the answers

    What does a standard deviation of 0.45 hr/day suggest about the children's time spent playing outdoors?

    <p>Outdoor time is relatively consistent.</p> Signup and view all the answers

    What is the z-score for a salary of $70,000?

    <p>2.25</p> Signup and view all the answers

    What area under the curve corresponds to a z-score of 2.25?

    <p>0.0122</p> Signup and view all the answers

    What is the probability of a salary being between $60,000 and $66,000?

    <p>0.4931</p> Signup and view all the answers

    What z-score corresponds to the 94th percentile of the salary distribution?

    <p>1.55</p> Signup and view all the answers

    If a graduate's salary is $67,200, how many standard deviations above the mean is this salary?

    <p>1.55</p> Signup and view all the answers

    Why is the z-score for the 94th percentile stated as being between 1.55 and 1.56?

    <p>It rounds the z-score up to two decimal places.</p> Signup and view all the answers

    The value $66,000 corresponds to which z-score?

    <p>1.25</p> Signup and view all the answers

    Which statement is true regarding the z-table?

    <p>It provides areas for both positive and negative z-values.</p> Signup and view all the answers

    What is the value of Q1 in the given distribution?

    <p>1.40</p> Signup and view all the answers

    What is the value of the IQR for the given data?

    <p>0.60</p> Signup and view all the answers

    How is the 80th percentile of the distribution calculated?

    <p>Using z = 0.84</p> Signup and view all the answers

    If the mean starting salary is $61,000, what is the z-score for a salary of $70,000?

    <p>1.25</p> Signup and view all the answers

    What is the probability that a randomly selected college graduate has a starting salary between $60,000 and $66,000?

    <p>0.50</p> Signup and view all the answers

    What is the z-score corresponding to the 94th percentile?

    <p>1.88</p> Signup and view all the answers

    To find a salary that is larger than 94% of the starting salaries, what calculation should be performed?

    <p>X = 61,000 + 1.88(4,000)</p> Signup and view all the answers

    What percentage of primary school age children spend more than 2.08 hrs/day playing outdoors?

    <p>20%</p> Signup and view all the answers

    Study Notes

    Normal Distribution

    • The Normal distribution is a continuous probability distribution that is bell-shaped and symmetrical.
    • The Normal distribution is often used to model real-world data, such as height, weight, and test scores.
    • The Normal distribution is determined by its mean (μ) and standard deviation (σ).
    • The mean of a Normal distribution is equal to the median.
    • The total area under the Normal curve is 1.
    • A Normal distribution with mean 0 and standard deviation 1 is called the Standard Normal distribution.

    Empirical Rule

    • The Empirical Rule states that approximately 68% of the observations in a Normal distribution lie within one standard deviation of the mean.
    • The Empirical Rule states that approximately 95% of the observations in a Normal distribution lie within two standard deviations of the mean.
    • The Empirical Rule states that approximately 99.7% (almost all) of the observations in a Normal distribution lie within three standard deviations of the mean.

    Z-Scores

    • A Z-score is a measure of how many standard deviations an observation is from the mean.
    • Z-scores can be used to compare observations from different Normal distributions.
    • Z-scores are calculated using the formula: z = (x - μ) / σ, where x is the observation, μ is the mean, and σ is the standard deviation.

    Calculating Probabilities and Percentiles

    • To calculate the probability of an observation falling within a certain range, you can use the Standard Normal probability table (z-table).
    • You can also calculate percentiles using the z-table.
    • The z-table gives the cumulative area to the left of a given z-score.
    • Percentiles are calculated by finding the z-score that corresponds to the desired percentile.
    • The Interquartile Range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1).

    Examples

    • The time spent playing outdoors among primary school age children is Normally distributed with a mean of 1.70 hours per day and a standard deviation of 0.45 hours per day.
    • Nearly all primary school age children spend between 0.35 and 3.05 hours per day playing outdoors.

    Using the Normal Distribution in Real World Examples

    • According to reports, starting salaries for college graduates in a certain city are Normally distributed with a mean of $61,000 and a standard deviation of $4,000.
    • Approximately 1.22% of college graduates have starting salaries greater than $70,000.
    • About 49% of new graduates have starting salaries between $60,000 and $66,000.
    • A graduate must earn at least $67,200 to be in the top 6% of earners.

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    Normal Distributions 2021 PDF

    Description

    Explore the concepts of Normal distribution and the Empirical Rule in this quiz. Learn how to apply these statistical principles to real-world data, such as height and test scores. Understand the significance of mean, standard deviation, and areas under the curve.

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