Statistics Problems

 Question #1 (10 marks):

OK, so I asked twenty people I met on the street in downtown Oshawa how old they were. They said they were:

	Raw Data (years)		Ranked Data
		Location
	45	1	18
	67	2	23
	24	3	24
	46	4	26
	76	5	27
	37	6	37
	56	7	43
	43	8	44
	27	9	44
	89	10	44
	44	11	45
	62	12	46
	18	13	56
	44	14	56
	78	15	62
	65	16	65
	23	17	67
	44	18	76
	56	19	78
	26	20	89
Sum	970		970

a) Determine three measures of central tendency for these data (6)

b) From the ranked data, determine the 90^th percentile (4)

2. (19 marks)

Using the data from the previous question (#1)

a)            Starting with a lower class limit of 10 years of age and a class width of 10 years, construct an Ogive of these data. (5)

b)            Determine the 90^th percentile from this graph (2),

c)            Are the two 90^th percentiles the same? Why or why not? (1,2)

d)            Determine the upper and lower outlier limits (1,1)

e)            If this sample is truly indicative / representative of the people in Oshawa, what percentage of people in Oshawa are drawing the Old Age Security cheques? (2)

f)             Construct the boxplot (five-number summary) describing these data. Using either the information from the raw data or the information from the Ogive. (5)

3. (6 marks)

For native Inuit children born in the 80s, the probability that a child's gestational age is less than 37 weeks is 0.185 and the probability that the child's birth weight is less than 2500 grams is 0.055. Furthermore, the probability that these two events occur simultaneously is 0.030. Let event A be the event that the infant's gestational age is less than 37 weeks and B be the event that his or her birth weight is less than 2500 grams.

a)    Construct the Venn diagram depicting these two events.(1)

b)    Are events A and B independent? (1)

c)    For a randomly selected Inuit baby, what is the probability that events A or B occurred, or both occurred? (2)

d)    What is the probability that event A occurred given that event B occurred? (2)

v4. 4. (10 marks)

The following data are taken from a study investigating the use of a technique called radionuclide ventriculography as a diagnostic test for detecting coronary artery disease.

Test	Disease		Total
	Present	Absent
Positive	302	80	382
Negative	179	372	551
Total	481	452	933

a)     What is the sensitivity of radionuclide ventriculography in this study? 

b)     What is the specificity of this test?     

c)     For a population in which the prevalence of coronary artery disease is 0.10, calculate the probability that an individual has the disease given that he or she tests positive using the radionuclide ventriculography test. (This is an application of Bayes Theorem)  (8)

Note: in your analysis, let T⁺ denote a positive test result, T^- denote a negative test result, D⁺ denote the patient has the disease and D^- denote the patient does not have the disease. 

5. (10 marks)
A survey of 500 top Canadian hospitals reported what the hospitals' hiring outlook was for the next 18 months as well as the general outlook for the economy over the same period of time.

	Hiring Outlook
Economy Outlook		Add Jobs	No Change	Cut Jobs	Totals
		(A1)	(A2)	(A3)
Favourable	(B1)	100	50	25	175
Unknown	(B2)	50	150	60	260
Unfavourable	(B3)	15	10	40	65
Totals		165	210	125	500

In the following, express your answers in the form: P(A_i) or P(B_i) to state your probability answer.         
a)  What is the probability that a randomly selected hospital from this sample does not have a favourable outlook for the economy over the next 18 months? (2)
b)  What is the probability that a randomly selected hospital from this sample has a favourable outlook for the economy and as a result, will be adding jobs over the next 18 months? (2)
c)    What is the probability that a randomly selected hospital plans to cut jobs during the next 18 months? (2)
d)    What is the probability that a randomly selected hospital does not plan to add jobs during the next 18 months?  (2)
e)    What is the probability that a randomly selected hospital has an unfavourable forecast for the economy or plans to cut jobs over the next 18 months? (2)

6. (8 marks)

In a recent edition of the CPR (Cardio-Pulmonary Resuscitation) handbook, the probability (survival rate) without any complications (for example, irreversible brain damage) based on time (minutes) of not breathing is as follows:

x (min)	P(x)
0	0.284
1	0.246
2	0.221
3	0.198
4	0.037
5	0.014
6	0+

a)            Is this distribution a "proper" probability distribution? Why? (1)
b)            Assuming that it is a proper probability distribution, determine the mean time when there is no discernible damage. (2)
c)            What is the standard deviation? (2)
d)            If there was a person who did not breathe for 4 minutes and that person did not suffer any brain damage, would that result appear to be unusual? Why? (1,2) 

 7. (7 marks)
The masses of packed saline from Company A follow a normal distribution with a mean of 1000 grams and a standard deviation of 50 grams.

a)            What percentage of the packed saline have masses less than 880 grams? (2)

b)            Ten per cent of the packed saline weigh more than what value? (2)

c)            Is it reasonable to conclude that Company A would pack a saline pack that would weigh more than 1085 grams? Explain. (1,2)

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