Solutions to Quiz 02

Notes:

  1. Note that you need to learn how to use your calculator. If you find that you cannot use your calculator well, then you have to go back to the basics.
  2. What I laid out here are recommended solutions. There could be other solutions.
  3. If you find yourself asking how many decimal places to use and there are no directions, my suggestion is to get a sense of how many decimal places would matter, especially given the units. For calculations leading to the final answer, it might be best not to round off “too much”. At the end of the day, your solution should not just be the final answers. Make sure to write down how you got to the final answers.
  4. You don’t have to draw the entire histogram for the questions here. You only need to draw the relevant rectangles. When you go over the solutions, DRAW THE RECTANGLES!
  5. Sometimes, you can do a quick check of whether you got correct answers. An example is Item 2a of the quiz. How would you check if you got the right answer?

For V26:

  1. The mean is \(\overline{x}=\dfrac{-1+3+5+7}{4}=3.5\). To get the standard deviation, calculate first \(\displaystyle\sum_{i=1}^n\left(x_i-\overline{x}\right)^2=(-1-3.5)^2+(3-3.5)^2+(5-3.5)^2+(7-3.5)^2=35\). Then \(s=\sqrt{35/3}\approx 3.42\).

    1. The total count is 65. The rectangle has a horizontal length of 19-16=3 and the area is 23/65. Therefore, the height of the rectangle is (23/65)/3 \(\approx\) 0.12. This means that the height of the rectangle is 12% per minute.
    2. The 75th percentile can be found in the interval 19-22 because (3+14+23)/65 \(\approx\) 0.62 and (3+14+23+12)/65 = 0.8. Focus on the rectangle over the interval 19-22. Its height is (12/65)/3. We need an additional area of about 0.75-0.62=0.13 to reach the 75th percentile. We need to know how much to add to 19 in order to reach an area of 0.13 for the rectangle over the interval 19-22. The horizontal length should be 0.13/((12/65)/3) \(\approx\) 2.11. So a value for the 75th percentile will be 19+2.11=21.11 minutes.

For V26:

  1. The mean is \(\overline{x}=\dfrac{-3-1+1+5}{4}=0.5\). To get the standard deviation, calculate first \(\displaystyle\sum_{i=1}^n\left(x_i-\overline{x}\right)^2=(-3-0.5)^2+(-1-0.5)^2+(1-0.5)^2+(5-0.5)^2=35\). Then \(s=\sqrt{35/3}\approx 3.42\).

    1. The total count is 80. The rectangle has a horizontal length of 7000-6000=1000 and the area is 17/80. Therefore, the height of the rectangle is (17/80)/1000 \(\approx\) 0.0002125. This means that the height of the rectangle is 0.02125% per dollar.
    2. The 25th percentile can be found in the interval 4000-5000 because (6+8)/80 =0.175 and (6+8+14)/80 = 0.35. Focus on the rectangle over the interval 4000-5000. Its height is (14/80)/1000. We need an additional area of about 0.25-0.175=0.075 to reach the 25th percentile. We need to know how much to add to 4000 in order to reach an area of 0.075 for the rectangle over the interval 4000-5000. The horizontal length should be 0.075/((14/80)/1000) \(\approx\) 428. So a value for the 25th percentile will be 4000+428=4428 dollars.