Reference no: EM132223661

Assignment - Critical Appraisal of Statistics in Health Science

Question 1: Organizing Data - Part A

With the legalization of Marijuana, I was interested in what is meant by "light" with respect to anything we inhale. So my focus turned to cigarettes; material for which we have all kinds of data. The declared concentrations of nicotine in milligrams (mg) per cigarette for forty different brands of Canadian cigarettes are shown in the following table. Some people believe nicotine levels less than 8 mg/cigarette are considered "light" - I don't!.

It is the nicotine in tobacco that is addictive. Each cigarette contains about 10 milligrams of nicotine. A person inhales only some of the smoke from a cigarette, and not all of each puff is absorbed in the lungs. The average person gets about 1 to 2 milligrams of the drug from each cigarette.

A "light" cigarette refers only to its nicotine content; not to the particulate content!

Nicotine (mg) per cigarette
5	9	12	13	8	14	9	12
4	10	1	15	2	16	14	14
23	7	15	19	14	13	16	14
19	5	16	6	3	16	10	14
27	8	18	13	16	21	6	3

a) Construct a frequency distribution table of these data and display the results as a histogram using a class width of 5 mg and starting at 0 mg.

b) How would you describe the shape of the histogram?

i. Support your decision by determining the three measures of central tendency. [note; sum = 480 mg]

ii. What would be the best measure of central tendency for these data? Why?

c) Construct the Ogive (Cumulative Less-than Frequency Distribution) for these data and present the results as a graph.

d) From the Ogive graph, determine/show the 25th, 50th and 75th percentile.

e) From the Ogive graph, what percentage (percentile) or how many of the cigarette brands would be considered "light"? Show this result on your Ogive.

f) From the ranked (raw) data, determine the 25th, 50th and 75th percentile. Are they the same values as found in part (d)? Why or why not?

g) Construct the box plot for these data.

h) Are there any "Outliers" in this data set? If so, what are they?  

Question 1 - Part B

From Wikipedia, an intelligence quotient (IQ) is a total score derived from several standardized tests designed to assess human intelligence. Let's assume that when current IQ tests were developed, the mean raw score of the norming sample is defined as IQ 100 and with a standard deviation (SD) of 15 IQ points.

Let's assume we conducted our own IQ test of students here at UOIT. We recruited 60 students and below is the resulting Cumulative Less-Than (Ogive) graph of their results.

From this graph

a) Determine Q₁, Q₂ and Q₃.

b) The top 10% of the students had an IQ greater than or equal to what value?

c) How many students had an IQ greater than or equal to 110 but less than 130 IQ points?

Question 2 - A Survival Analysis

A new antibiotic is being developed and the company that's developing this new drug wants to compare it to that of penicillin in the treatment of streptococcal pharyngitis (strep throat).

a) Several streptococcal pharyngitis bacterial colonies were developed for testing this new drug. After five days, the bacterial colonies treated by this new drug had decreased by 75%. In other words, if one started with 1000 bacterium, after 5 days, only 250 would have survived. An identical set of bacterial colonies treated by penicillin had decreased by 70% over the same time period; in other words, if one started with 1,000 bacterium and used penicillin, then 300 would be alive at the end of the 5 days.

From a survival analysis perspective, how efficient is the new antibiotic when compared to penicillin? In other words, what is the overall average percent decrease per day (or the survival rate per day) of these bacterial colonies for the new drug and that of penicillin?

As a check, the average colony strengths for each of these antibiotics were determined for each of the five days. The results were as follows:

	Percent Loss
	(When compared to the previous day's results)
	End of Day 1	End of Day 2	End of Day 3	End of Day 4	End of Day 5
New Antibiotic	35.0%	30.0%	25.0%	20.0%	8.4%
Penicillin	30.0%	40.0%	20.0%	10.0%	0.9%

b) What is the average per day decrease (in terms of percentage) or survival (either one) of the streptococcal pharyngitis bacteria for these two drugs? (Hopefully your results of part (b) compare favorably with that of part (a)).

c) How would you interpret these results? In other words, would you consider that the differences between these two drugs significant or meaningful! (I know we haven't covered this in any great detail yet, but just wanted to know what you would tell the big bosses of the company based on this small study.)

Question 3 -

i) The first card selected from a standard deck of cards was a king.

a) If it is returned to the deck, what is the probability that a king will be drawn on the second selection?

b) If it is not replaced, what is the probability that a king will be drawn on the second selection?

c) What is the probability that a king will be drawn on the first draw from the deck and another king on the second draw (assuming the first king was not returned)?

d) What is the probability of selecting the King of Hearts from a standard deck?

e) This probability is an example of (what type)_______________________ probability.

ii) Assume that there is a 15% probability of any shot Wayne Gretzky makes will result in a goal.

From a binominal perspective (either he will score or he will not), what is the probability that

a) In any game Wayne played, if he misses his first shot on goal, what is the probability that he will make his second shot?

b) If Wayne takes, on average, 280 shots on goal per season, how many goals is he expected to make for that season?

c) Is it unusual for Wayne to make 50 goals per season? Why (show your reasoning)?

iii) Now let's look at basketball and one of the most infamous free throw (FT) fouls shooters there ever was... according to the Bleacher report, Shaquille O'Neil made the top ten worst foul shooters in history (he was number four!). His career FT percentage was 52.7% and he took 11,252 free throws during his career with the Magic and the Lakers.

Many teams would play "Hack a Shaq" but that's another story.

So for this question, let's assume Shaq has to make the last three free throws in this particular game (he was fouled while taking a 3-pointer ???? ) and for this game, his shooting percentage is that staggering 53%.

a) If he makes the first free throw, what is the probability that he will make the second?

b) All three free throws are good.

c) None of the three free throws are good.

d) At least one of the free throws is good.

e) This probability is an example of __________________________ probability.

iv) OK the summer-time sport in Toronto is baseball; the Blue Joys. I kept some stats on the first 800 batters that went to-the-plate. The results are as follows:

	At the Plate - Results
Batter	Hits (H)	Strike Outs (SO)	Walks (W)	Total
Right Handed (R)	115	334	51	500
Left Handed (L)	145	456	99	700
Switch (S)	40	110	50	200
Totals	300	900	200	1400

a) So if the next Blue Joy batter that comes up is left handed, what is the probability that this batter will get on base? Determine probability P(H or W | L).

b) Since I don't know the players, what is the probability that the next batter that comes up to bat is a "Switch" hitter? This is the marginal probability, P(S).

c) Again, since I don't know the players, what is the probability that the next batter that comes up to bat will get a hit? Again, this is the marginal probability, P(H).

d) And once more, since I don't know the players, what is the probability that the next batter that comes up to bat will either be a right handed batter or a switch hitter? Again, this is a marginal probability, P(R or S).

e) What is the probability that the next two players that come to the plate will get hits? i.e., P( H₁ and H₂).

f) What is the probability that three batters in-a-row will not get a hit (either strike out or get a walk)? (I will let you think about this one on your own).

g) What is the probability that the next batter that comes up to the plate will strike out given that he is a switch hitter? i.e., P( SO | S )?

h) What is the probability that the next batter that comes up to the plate will be right handed given that he walked? i.e, P( R | W )?

i) What is this table called?

v) Two hundred students graduated five years ago from UOIT in the Health Science faculty. I attended their 5-year reunion and was amazed by their career choices. Of the original two hundred, 60 were "Bed Side" nurses; 40 were "Critical Care" nurses; and 30 were "Acute Care" nurses. Ten nurses were both "Critical Care" and "Acute Care" nurses.

a) So, what is the probability of randomly selecting a graduate at random and that graduate is not a "Bed Side" nor a "Critical Care" nor an "Acute Care" nurse.

b) Complete the following Venn diagram showing these probabilities.

vi) On a very cold winter's day, 20 percent of the support staff at Rouge Valley are absent from work. As a result, the support staff employees are selected at random for a special in-depth study on absenteeism (to find the reason for the high absenteeism). What is the probability of selecting 12 support staff employees at random on a cold winter's day and finding that none of them are absent?

vii) A sales representative calls on six hospitals in York Region. It is immaterial what order s/he calls on them. How many ways can s/he organize her/his calls?

viii) At a nearby hospital, the board of directors consists of eight men and six women. A three-member search committee is to be chosen at random to recommend a new CFO. What is the probability that all three members of the search committee will be women?

ix) Four defective electric toothbrushes were accidentally shipped to a drugstore by the manufacturer along with 12 non-defective ones. What is the probability that the first two electric toothbrushes sold will be returned to the drugstore because they are defective?

x) The Rouge Valley Health S¬¬¬ystem reports that 50 percent of its long-term-care residents are covered by the Great-West Life Assurance plan, 30 percent are covered by the General Motors plan, and 20 percent have both plans.

a) If a resident is chosen at random, what is the probability this resident has either a Great-West Life Assurance plan or the General Motors plan?

b) What is the probability the resident does not have either a Great West Assurance or the General Motors plan?

Question 4 - Bayes Theorem

According to a recent stats posting for 2014, the HIV prevalence rate in Canada is 212 cases per 100,000 population or 0.00212. A new screening test / procedure was set up by Healthie Canada and according to their research of 4,600 patients, the performance data / information is shown in the following table.

Have HIV? (Independent Variable)	Test Results (Dependent Variable)
	Positive	Negative	Total
Yes	321	3	324
No	1	4,275	4,276
	322	4,278	4,600

From this information, determine the following:

a) What is the probability that a patient who has HIV will test positive? i.e., P(positive ¦yes) = ?

b) What is the probability that a patient who does not have HIV will test positive? i.e., P(positive ¦no) = ?

c) What is the probability that a patient who does not have HIV will test negative? i.e., P(negative ¦no) = ?

d) So now, Mr Smith, a Canadian, has just tested positive. What is the probability that he actually has the disease? i.e., P(yes ¦ positive) = ?

Question 5 -

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	0.351
1	0.238
2	0.224
3	0.142
4	0.037
5	0.008
6	0+

a) Is this distribution a "proper" probability distribution? Why? (looking for two reasons).

b) Assuming that it is a proper probability distribution, determine the mean time when there is no discernible damage.

c) What is the standard deviation?

d) And again, assuming that it is a proper probability distribution, construct the probability graph depicting same.

e) If there was a person who did not breathe for 4 minutes and that person did not suffer any brain damage, would that result be considered unusual? Why?

Question 6 -

You are at Tim Ortons and you note the time of the following 8 people standing in line.

	Time (Seconds)
	119
	116
	143
	131
	145
	126
	160
	116
Sum	1,056

From this sample of eight coffee drinkers

a) Determine the Mean wait time.

b) Determine the Standard Deviation.

c) What is the mean wait time for all coffee drinkers at this Tim Ortons?

Believe it or not, the indoor (not drive-through) wait times for a different Tim Orton's coffee is normally distributed with a mean of 2.35 minutes and a standard deviation of 45 seconds. If it gets any longer than this, they open up a new wait line. These guys are good!

So you arrive unannounced

i. What is the probability that you will have to wait less than 1 minute for your coffee?

ii. What is the probability that you will have to wait longer than 4 minutes for your coffee?

iii. Construct the pdf showing parts i) and ii).

Now assume that the class has just ended and you as a group of 36 students approach this Tim Orton's.

iv. What is the probability that the mean wait time for your group to receive their coffee is less than 2 minutes?

v. What is the probability that the mean wait time for your group to receive their coffee is greater than or equal to 2 minutes and less than or equal to 2 minutes 30 seconds?

vi. What is the probability that the mean wait time for your group to receive their coffee is greater than 2 minutes 30 seconds?

vii. Construct the pdf showing parts iv), v) and vi).

Question 7 -

i) A sample of 64 elements was randomly extracted from a population. This sample's mean was 240 units and its standard deviation was 14 units.

a) Develop the 80% confidence interval for the unknown population mean.

b) Develop the 99% confidence interval for the unknown population mean.

c) What is the obvious difference in these two confidence intervals?

ii) At the World Championship Dart contest, bets were being placed on the accuracy of the contestants. One of the braggarts said his accuracy was only 80% but he wanted to play with the "Big Boys". So they gave him three darts and the object of the betting contest was to see how many of these three darts he could stuff into the Bull's Eye. So if his accuracy was only 80%, then determine from first principles (not using the Binominal chart or the Binominal formula but using the Union, Complement, Intersection and Combination rules) what would be the betting line (probabilities) for the following scenarios.

a) What is the probability that he could get all three darts in the Bull's eye?

b) What is the probability that he could get two of the three darts in the Bull's eye?

c) What is the probability that he could get one of the three darts in the Bull's eye?

d) What is the probability that none of his darts found the Bull's eye?

e) What is the probability that at least one of his darts found the Bull's eye?

iii) It is assumed that 66% of the population is in support of the new Sex Education program proposed by the new Provincial Government. A large sample of 205 people were asked whether they supported it or not. (A simple question.)

What is the probability of randomly recruiting a large sample of this size and getting a result greater than 70% in support of this new program? (i.e.greater than 143 people in this sample saying yes! )

In answering this question, prepare the PDF.

iv) Over 3000 students have taken this course over the past 12 years. The mean (average) mark for these students is 72% and the standard deviation is 12%. Is it unusual for a student to get a mark greater than 90% (an A+) in this course?

Also prepare the PDF depicting this problem.

Attachment:- Assignment File.rar