Q1: A density curve is always plotted above which of the following? (a) The x-axis (b) The y-axis (c) The mean (d) The median
Solution:
Ans: (a) Explanation: A density curve must always lie on or above the x-axis because it represents probabilities or proportions, which cannot be negative. The curve describes the overall pattern of a distribution.
Q2: What is the total area under any density curve? (a) 0 (b) 0.5 (c) 1 (d) 100
Solution:
Ans: (c) Explanation: The total area under a density curve is always exactly 1 (or 100%). This represents the fact that the probability of all possible outcomes must equal 1.
Q3: In a symmetric density curve, which relationship is true? (a) Mean > Median (b) Mean <> (c) Mean = Median (d) Mean and Median cannot be determined
Solution:
Ans: (c) Explanation: In a symmetric density curve, the mean and median are equal and located at the center of the distribution. The curve is balanced around this central point.
Q4: For a right-skewed density curve, which statement is correct? (a) Mean < median=""><> (b) Mode < median=""><> (c) Median < mode=""><> (d) Mean = Median = Mode
Solution:
Ans: (b) Explanation: In a right-skewed distribution, the tail extends to the right. The mean is pulled in the direction of the tail (right), so Mode < median="">< mean.="" the="" mean="" is="" most="" affected="" by="" extreme="">
Q5: A uniform density curve on the interval from 2 to 8 has what height? (a) \(\frac{1}{10}\) (b) \(\frac{1}{8}\) (c) \(\frac{1}{6}\) (d) \(\frac{1}{4}\)
Solution:
Ans: (c) Explanation: For a uniform density curve, the height is calculated as \(\frac{1}{\text{width}}\). The width is \(8 - 2 = 6\), so the height is \(\frac{1}{6}\). This ensures the total area equals 1.
Q6: What does the area under a density curve between two values represent? (a) The frequency of observations (b) The proportion of observations (c) The total number of data points (d) The range of the data
Solution:
Ans: (b) Explanation: The area under a density curve between two values represents the proportion (or probability) of observations falling in that interval. It is not a count but a fraction of the total.
Q7: The median of a density curve divides the area under the curve into what parts? (a) Three equal parts (b) Four equal parts (c) Two equal parts of 0.5 each (d) Two parts of 0.25 and 0.75
Solution:
Ans: (c) Explanation: The median is the point that divides the area under the density curve into two equal parts, each with area 0.5. Half of the observations fall below the median and half above.
Q8: For a left-skewed density curve, where is the mean located relative to the median? (a) To the right of the median (b) To the left of the median (c) At the same location as the median (d) At the mode
Solution:
Ans: (b) Explanation: In a left-skewed distribution, the tail extends to the left. The mean is pulled toward the tail (left), so it is located to the left of the median.
## Section B: Fill in the Blanks
Q9: A density curve is an idealized description of a distribution that smooths out the irregularities in a __________.
Solution:
Ans: histogram Explanation: A density curve is a smooth mathematical model that represents the overall pattern of a distribution, replacing the bars of a histogram with a continuous curve.
Q10: The balance point of a density curve, where the curve would balance if made of solid material, is called the __________.
Solution:
Ans: mean Explanation: The mean of a density curve is located at the balance point of the distribution. This is the point where the curve would balance if it were a physical object.
Q11: A density curve that has the same shape on both sides of its center is called __________.
Solution:
Ans: symmetric Explanation: A symmetric density curve has mirror-image halves. The left and right sides of the center are identical in shape.
Q12: In a uniform distribution, all intervals of the same length have __________ probability.
Solution:
Ans: equal Explanation: A uniform distribution is characterized by constant probability density across its range. All intervals of equal width have the same probability.
Q13: The point in a density curve where exactly 25% of the area lies to its left is called the __________ quartile.
Solution:
Ans: first Explanation: The first quartile (Q1) is the value below which 25% of the distribution lies. It marks the 25th percentile of the data.
Q14: A density curve with a long tail extending to the right is described as __________ skewed.
Solution:
Ans: right Explanation: A right-skewed (or positively skewed) distribution has a tail that extends toward higher values on the right side of the distribution.
## Section C: Word Problems
Q15: A density curve is defined by the equation \(y = 0.2\) on the interval from 0 to 5, and \(y = 0\) elsewhere. Find the probability that a randomly selected value falls between 1 and 3.
Solution:
Ans: Step 1: This is a uniform density curve with height \(y = 0.2\) from 0 to 5. Step 2: The probability is the area of the rectangle with width \(3 - 1 = 2\) and height 0.2. Step 3: Area = width × height = \(2 \times 0.2 = 0.4\) Final Answer: 0.4 or 40%
Q16: A triangular density curve has a base from 0 to 10 and reaches its peak at \(x = 5\) with a height of 0.2. What is the total area under this density curve?
Solution:
Ans: Step 1: The area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\). Step 2: Base = \(10 - 0 = 10\) and height = 0.2. Step 3: Area = \(\frac{1}{2} \times 10 \times 0.2 = 1\) Final Answer: 1
Q17: The waiting time for a bus is uniformly distributed between 0 and 20 minutes. What is the probability that a passenger waits between 5 and 12 minutes?
Solution:
Ans: Step 1: For a uniform distribution from 0 to 20, the height of the density curve is \(\frac{1}{20} = 0.05\). Step 2: The width of the interval is \(12 - 5 = 7\) minutes. Step 3: Probability = width × height = \(7 \times 0.05 = 0.35\) Final Answer: 0.35 or 35%
Q18: A density curve is uniform on the interval [4, 14]. What is the height of this density curve?
Solution:
Ans: Step 1: For a uniform density curve, height = \(\frac{1}{\text{width}}\). Step 2: Width = \(14 - 4 = 10\). Step 3: Height = \(\frac{1}{10} = 0.1\) Final Answer: 0.1
Q19: A density curve has the shape of a rectangle from \(x = 2\) to \(x = 8\). If the total area under the curve must equal 1, find the height of the rectangle.
Solution:
Ans: Step 1: Area of rectangle = width × height. Step 2: Width = \(8 - 2 = 6\). Step 3: Since area must equal 1: \(6 \times h = 1\), so \(h = \frac{1}{6}\). Final Answer: \(\frac{1}{6}\) or approximately 0.167
Q20: The median of a continuous distribution represented by a density curve is 45, and the area under the curve to the left of 45 is 0.5. What is the area under the curve to the right of 45?
Solution:
Ans: Step 1: The total area under any density curve equals 1. Step 2: The median divides the distribution so that 0.5 of the area is to the left. Step 3: Area to the right = Total area - Area to the left = \(1 - 0.5 = 0.5\). Final Answer: 0.5
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