35 53 61 31 21 42 23 29 35 37 39 58 27 64 27 31 36 48 41 22 37 35 42 32 43 34 59 50 38 43 31 30 41 37 29 45 23 56 46 41
- (i) Obviously, 35% of the blood donors who visit
the bloodmobile at this company this particular day fall into the
30 to 39 years age group. Is this number (i.e.,
35%) a parameter or a statistic?
- (ii) Based on this data, do you think it is correct to state that
about 35% of all
blood donors anywhere in the United States fall into the 30 to 39 years
age group? Explain, using the appropriate statistical vocabulary.
- (iii) A local newspaper reporter from the Utah Statesman wants to learn more
about some of these blood donors and their motivation to come to
the bloodmobile. Since the reporter once took a Stat 2000 class,
she thinks that it might be best to randomly select people for
the interview rather than taking the first few people that showed
up at the bloodmobile. The reporter wants to talk to 5 people
and the bloodmobile operator hands her a list of those 40 people
that participated.
Explain how you would proceed if you were the reporter, using the appropriate statistical vocabulary. Use the random digits table (Table B), starting at line 120, to draw your sample. Which persons (indicate the numbers and circle the corresponding age on the previous page) do you finally select for your interview?
- (iv) Based on your 5 persons from (iii) above, calculate
the sample proportion ``p-hat'' of blood donors at this company
that falls into the 30 to 39 years age group. Is this number a
parameter or a statistic?
- (v) Unfortunately, our reporter does not have the time
to calculate the mean age for those 5 persons from (iii) above.
Instead, she asks her assistant (a student who dropped out
of Stat 2000 after the first week) to calculate the mean age
of these 5 persons. The student comes up with 22 years.
Should the reporter be suspicious about this number or
is this just a possible effect of sampling? Explain,
using the appropriate statistical vocabulary. Note that
the reporter may be using different persons than the 5 persons
you used in the previous parts of this question.
Variant 1: Use the random digits table B and assume that the 2-digit numbers represent the age (we pretend that we know the ages of all blood donors in our population of size 40). If a 2-digit random number represents a valid age that exists in the population, we include the first person of this age into our sample. If the same 2-digit random number occurs again, we include the second person of this age into our sample. For example, in our population, the age 37 occurs three times, the age 23 occurs two times, etc. On the first occurrance of a 37 in the random digit table, the first person of age 37 will be included. On the second orrurrance the second person. We continue until we have a sample of size n.
Variant 2: Similar to Variant 1. However, if an age occurs more then once in our population, we would use another random digit to randomly select the person among all persons that are of the same previously determined age. The person selected in this 2-step procedure will be included into our sample. We continue until we have a sample of size n.
Starting at line 101, we would end up with ages 19 (not in population), 22 (include this person), 39 (include this person), 50 (include this person), 34 (include this person), etc.
The big question now is - which of these variants represent a simple random sample? Explain! To support your reasoning, you should assume that we just have a population of size 3 of ages 22, 37, and 37, i.e., person 1: 22, person 2: 37, and person 3: 37. We are interested in a SRS of size 2. Indicate the probabilities to end up with a sample that contains persons 1 & 2, persons 1 & 3, or persons 2 & 3 under Variant 1 and Variant 2, respectively. So, which variant represents a SRS? (5 Points)
Based on your answers above, how likely is an outcome of an average age of 23.2 years? (3 Points)
Based on your previous results, indicate the cumulative distribution function F(y) [this function has to be defined separately over 4 different intervals] and draw a graph of F(y). Does this graph fulfill the conditions for a cdf we listed in class?
Hint: To be able to determine the area under a triangle, we have to determine its width and height. This can be done by using the fact that each border line segment of a triangle can be decribed by a straight line. Determine functions y = a x + b for these line segments first, then determine the area under these line segments. All you need is simple geometry here! (5 Points)