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Based on what you have studied in this unit, reflect and answer the following questions:What are the main characteristics of the normal distribution?Why is it advantageous to use the Z scores?For a standard normal distribution, about 68% of values falls within one standard deviation of the mean; about 95% falls within two standard deviations; and about 99.7% falls within three standard deviations. Suppose the distribution of math scores in the SAT follow a normal distribution with a mean µ=700 and a standard deviation σ=150.Calculate the Z-score for an SAT Math score of 600. Interpret it in your own word.Find the interval [a, b] that captures 68% of the SAT Math score in this example.Find the interval [a, b] that captures 95% of the SAT Math score in this example.Find the interval [a, b] that captures 99.7% of the SAT Math score in this example.

Question

Based on what you have studied in this unit, reflect and answer the following questions:What are the main characteristics of the normal distribution?Why is it advantageous to use the Z scores?For a standard normal distribution, about 68% of values falls within one standard deviation of the mean; about 95% falls within two standard deviations; and about 99.7% falls within three standard deviations. Suppose the distribution of math scores in the SAT follow a normal distribution with a mean µ=700 and a standard deviation σ=150.Calculate the Z-score for an SAT Math score of 600. Interpret it in your own word.Find the interval [a, b] that captures 68% of the SAT Math score in this example.Find the interval [a, b] that captures 95% of the SAT Math score in this example.Find the interval [a, b] that captures 99.7% of the SAT Math score in this example.

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Solution

  1. The main characteristics of the normal distribution are:

    • It is symmetric: The left half of the distribution is a mirror image of the right half.
    • The mean, median, and mode are all equal and are located at the center of the distribution.
    • It is unimodal: It has one peak.
    • The distribution is defined by two parameters: the mean (µ) and the standard deviation (σ).
    • The total area under the curve is 1.

  2. Z-scores are advantageous because they allow us to compare scores from different normal distributions. They tell us how many standard deviations away a score is from the mean. This standardizes the score and allows for comparison across different scales.

  3. To calculate the Z-score for an SAT Math score of 600, we use the formula Z = (X - µ) / σ. Here, X is the score, µ is the mean, and σ is the standard deviation. So, Z = (600 - 700) / 150 = -0.67. This means that the score of 600 is 0.67 standard deviations below the mean.

  4. For a standard normal distribution, 68% of values fall within one standard deviation of the mean. So, the interval that captures 68% of the SAT Math score is [µ - σ, µ + σ] = [700 - 150, 700 + 150] = [550, 850].

  5. Similarly, 95% of values fall within two standard deviations of the mean. So, the interval that captures 95% of the SAT Math score is [µ - 2σ, µ + 2σ] = [700 - 2150, 700 + 2150] = [400, 1000].

  6. Finally, 99.7% of values fall within three standard deviations of the mean. So, the interval that captures 99.7% of the SAT Math score is [µ - 3σ, µ + 3σ] = [700 - 3150, 700 + 3150] = [250, 1150].

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