Density curves and normal distributions/Self-check assessment

Use the following quiz questions to check your understanding of density curves and normal distributions. Note that as soon as you have indicated your response, the question is scored and feedback is provided. As feedback is provided for each option, you may find it useful to try all of the responses (both correct and incorrect) to read the feedback, as a way to better understand the concept.

Density curves

Density curves

• Which of the following statistics is the balance point of a density curve, the point at which the curve would balance if made of solid material?
  • Median
    • That's not quite right. The median of a density curve is the point that divides the area under the curve into halves, an upper half and a lower half. Try again.
  • IQR
    • That's not quite right. The IQR is the middle 50% of the area under the curve. The balance point relates to the hypothetical weight of the curve, considering each observation as a weight. Try again.
  • Mean
    • That's correct. The mean of a density curve may be considered the balance point.
  • Standard deviation
  • That's not quite right. The standard deviation is not visually apparent on many density curves. Try again.

Density curves

• The full area under a density curve is 1
  • That's correct. The area is 1 by definition, meaning that the probability that a score chosen at random will occur under the curve is 1.
  • That's not quite right. By definition the area under a density curve represents 100% of the data. Hint: what proportion is equivalent to 100%? Try again.

Normal distributions

Normal distributions

• Select all of the statements that are true about a Normal distribution.
  • It is symmetric around its mean.
    • That's correct. A Normal distribution is a symmetric, unimodal, bell-shaped density curve.
  • The mean, median, and mode are equal.
    • That's correct. The mean, median and mode are equal for a Normal distribution.
  • It is defined by its mean and skew.
    • That's not quite right. A Normal distribution is defined by its mean and standard deviation.
  • The standard deviation is the distance from the center of the curve to the point on each side at which the curve changes from falling ever more steeply to falling ever less steeply.
    • That's correct. The standard deviation is visually apparent on a Normal distribution, using the points at which the curvature changes on either side of the center.
  • It has high density in the upper and lower tails.
    • That's not quite right. In Normal distributions, the highest density occurs in the central region.
  • Its mean and standard deviation are denoted by [math]\bar{x}[/math] and sd.
    • That's not quite right. The notation [math]\bar{x}[/math] and sd are used with actual observations. We use μ and σ to denote the mean and standard deviation of a density curve, such as the Normal distribution.
• A Normal or bell-shaped distribution has its greatest probability density in its tails.
  • True
    • That's not quite right. The distribution is higher and therefore denser in the middle of the distribution.
  • False
    • That's correct. The distribution is higher and therefore denser in the middle of the distribution.

Weight is a measure that tends to be Normally distributed. Suppose the mean weight of all women at a large university is 135 pounds, with a standard deviation of 12 pounds. Use the 68-95-99.7 standard deviation rule to answer the following two questions. (It may be useful to draw the curve and apply the sd rule for this distribution.)

• Which of the following ranges would include 99.7% (nearly all) of the women's weights?
  • above 135
    • That's not quite right. 135 is the mean weight, so approximately 50% of the women weigh more. Try again.
  • below 135
    • That's not quite right. 135 is the mean weight, so approximately 50% of the women weigh less. Try again.
  • between 123 and 147
    • That's not quite right. Given the standard deviation is 12, 123 and 147 are exactly one standard deviation below and above the mean, 135. The standard deviation rule tells us that only 68% of women's weights would be included in this range. Try again.
  • between 111 and 159
    • That's not quite right. Given the standard deviation is 12, 111 and 159 are exactly two standard deviations below and above the mean, 135. The standard deviation rule tells us that only 95% of women's weights would be included in this range. Try again.
  • between 99 and 171
    • That's correct. The 68-95-99.7 standard deviation rule tells us that virtually all the data fall within 3 standard deviations of the mean, which in this case is exactly between 135 - 3(12) = 99 and 135 + 3(12) = 171.
• What percentage of women weigh less than 123?
  • 50%
    • That's not quite right. According to the 68-95-99.7 standard deviation rule, 68% of women's weights fall between 135 - 12 = 123 and 135 + 12 = 147 pounds, which means that the remaining 32% of women's weights are divided evenly between less than 123 pounds and more than 147 pounds. Try again.
  • 100%
    • That's not quite right. According to the 68-95-99.7 standard deviation rule, 68% of women's weights fall between 135 - 12 = 123 and 135 + 12 = 147 pounds, which means that the remaining 32% of women's weights are divided evenly between less than 123 pounds and more than 147 pounds. Try again.
  • 2.5%
    • That's not quite right. According to the 68-95-99.7 standard deviation rule, 68% of women's weights fall between 135 - 12 = 123 and 135 + 12 = 147 pounds, which means that the remaining 32% of women's weights are divided evenly between less than 123 pounds and more than 147 pounds. Try again.
  • 16%
    • That's correct. According to the 68-95-99.7 standard deviation rule, 68% of women's weights fall between 135 - 12 = 123 and 135 + 12 = 147 pounds, which means that the remaining 32% of women's weights are divided evenly between less than 123 pounds and more than 147 pounds. Half of 32% is 16%.
  • 68%
    • • • • That's not quite right. According to the 68-95-99.7 standard deviation rule, 68% of women's weights fall between 135 - 12 = 123 and 135 + 12 = 147 pounds, which means that the remaining 32% of women's weights are divided evenly between less than 123 pounds and more than 147 pounds. Try again.