96, 92, 92, 89, 88, 87, 87, 87, 87, 80, 79, 79, 78, 76, 76, 76, 76, 74, 73, 72, 72, 71, 71, 71, 70, 66, 66, 60, 53, and 20. Find the mean, median, mode, and standard deviation. If this class is graded "on a curve," prepare a grading scale.
b. Find the mean, median, mode, range, and standard deviation of the ages of the actors winning the best actor award at the Academy Awards. See Table 10.3.
G45. Toss a toothpick onto a hardwood floor 1,000 times as described in Figure G.8, or toss 1,000 toothpicks, one at a time, onto the floor. Let be the length of the toothpick and d be the distance between the parallel lines determined by the floorboards.
Figure G.8 Buffon's needle problem
Equipment needed: A box of toothpicks (of uniform length) and a large sheet of paper with equidistant parallel lines. A hardwood floor works very well instead of using a sheet of paper. The length of a toothpick should be less than the perpendicular distance between the parallel lines.
- Guess the probability p that a toothpick will cross a line. Do this before you begin the experiment. The members of your group should reach a consensus before continuing.
- Perform the experiment and find p empirically. That is, to find p, divide the number of toothpicks crossing a line by the number of toothpicks tossed (1,000 in this case).
- By direct measurement, find and d.
- Calculate 2 and pd, and (2)/(pd).
- Formulate a conclusion. This is an experiment known as Buffon's needle problem.
G46. You are interested in knowing the number and ages of children (0-18 years) in a part (or all) of your community. You will need to sample 50 families, finding the number of children in each family and the age of each child. It is important that you select the 50 families at random. How to do this is a subject of a course in statistics. For this problem, however, follow these steps:
Step 1. Determine the geographic boundaries of the area with which you are concerned.
Step 2. Consider various methods for selecting the families at random. For example, could you:
(i)select the first 50 homes at which someone is at home when you call?
(ii)select 50 numbers from a phone book that covers the same geographic boundaries as those described in step 1?
(iii) Using (i) or (ii) could result in a biased sample. Can you guess why this might be true? In a statistics course, you might explore other ways of selecting the homes. For this problem, use one of these methods.
Step 3. Consider different ways of asking the question. Can the way the family is approached affect the response?
Step 4. Gather your data.
Step 5 .Organize your data. Construct a frequency distribution for the children, with integral values from 0 to 18.
Step 6. Find out the number of families who actually live in the area you've selected. If you can't do this, assume that the area has 1,000 families.
- What is the average number of children per family?
- What percent of the children are in the first grade (age 6)? If all the children aged 12-15 are in junior high, how many are in junior high for the geographic area you are considering?
- See if you can actually find out the answers to parts b and c, and compare these answers with your projections.
- What other inferences can you make from your data?