1. The mercury content in arctic char (a type of fish) is a health concern for those who eat it. A survey randomly samples 200 of these fish (from various areas) to determine the mean mercury content in arctic char. Unknown to the researchers, the mean mercury level in arctic char is 1.2 mmol/L with a standard deviation of 2.3 mmol/L. The sample mean they obtain is 1.3 mmol/L.
a. What is the population they wish to study?
b. What is the population parameter the researchers wish to study?
c. What is the probability that another such sample of size 200 would lead to a sample mean which is higher than that of the present sample?
2. A geneticist is growing many plants to obtain some that have a specific mutation. Radiation on the seeds is used to increase the probability of the mutation. Through this process, 32% of plants develop the mutation. She applies this process to 75 plants, the sample proportion of plants that will develop the mutation has a 95% probability of lying between _______ % and ________ %.
3. Antidepressants and nicotine patches are often used to help smokers quit. In a smoking cessation program involving a very large number of smokers, 70% of smokers use nicotine patches, 50% use antidepressants, and 20% use both nicotine patches and antidepressants. Of all nicotine patch users, 10% successfully quit smoking.
a. A smoker joining the program is randomly chosen.
i. Find the probability that the smoker uses antidepressants given that he uses nicotine patches.
ii. Find the probability that the smoker uses nicotine patches only.
iii. Are using nicotine patches and using antidepressants independent events? Justify your answer.
b. Five smokers who use nicotine patches are chosen with replacement. Find the probability that at least one quits smoking successfully.
c. The program manager wants to compare the effectiveness of antidepressants and nicotine patches in smoking cessation. For the incoming 100 program participants, he plans to randomize them to one of the four treatments:
- Treatment 1: placebo antidepressant and placebo nicotine patch
- Treatment 2: placebo antidepressant and nicotine patch
- Treatment 3: antidepressant and placebo nicotine patch
- Treatment 4: antidepressant and nicotine patch
The individuals are followed to see if they successfully quit smoking at the end of the program.
i. How many factor(s) is / are under consideration? List the factor(s).
ii. What is the response variable?
iii. Why do you think placebo antidepressant / nicotine patch are needed?
iv. Which of the following is the most appropriate graphical display for displaying the data from the different treatments? Circle only one answer.
A. Side-by-side boxplots.
B. A bar graph.
C. A stemplot.
D. A histogram.
4. Some smokers who wear nicotine patches for long hours experience abnormally vivid dreams and nightmares (especially when they wear the patches while they sleep). Eighty-one individuals who have this side effect are interviewed. Their average daily numbers of hours of wearing nicotine patches and frequency of nightmares in the past month are recorded.
d. The following shows a boxplot of the frequency of nightmares these smokers experienced in the past month.
What can you say about the distribution of the frequency of nightmares? Check all that apply. [3 marks]
____ The distribution is bell-shaped.
____ The distribution is skewed to the left.
____ The distribution is skewed to the right.
____ The maximum number of nightmares experienced in the past month is 24.
____ More than 25% of the smokers had more than 10 nightmares in the past month.
____ The IQR of the number of nightmares experienced in the past month is about 20.
e. The following shows a scatterplot of the frequency of nightmares in the past month versus the average daily number of hours of wearing the nicotine patches.
i. A smoker wore the patch 24 hours a day and he had a total of 40 nightmares last month. His observation is (circle all that apply) [2 marks]
an x-outlier a y-outlier a model outlier none of these
ii. Consider the data with the observation in (i) removed. There are now 80 observations, and summary statistics are given below.
|
Variable
|
Mean
|
Standard deviation
|
|
Number of hours of wearing patches
|
20.48
|
1.90
|
|
Frequency of nightmares
|
9.14
|
5.90
|
The correlation coe?cient between the two variables is 0.86.
(1) Predict the frequency of nightmare experienced in a month when a
smoker wears the patch for 18 hours per day. Show your calculations.
(2) A smoker in the data set wears nicotine patches 21 hours a day. He
wants to see how accurate the regression line predicts the number of nightmares he experienced. He calculates that the residual for his number of nightmares is 10. Explain why he must have made a mistake in his calculation of the residual. Use the scatterplot to answer this question.
(3) The correlation coefficient before removing the observation in (i) is
(circle the most appropriate answer)
-0.56 0.05 0.82 0.90
5. Gun purchasing has never been higher in the USA than since the last federal election. A grass-root organization interested in fighting for more gun control is collecting data to better understand the situation. They wish to know the percentage of gun owners that own them for home protection. Not having a list of all gun owners, they get a list of all shooting ranges and randomly select 3. From the 3 shooting ranges they question all the members.
f. For the sampling described above, check all that apply
_____ The target population is all American members of shooting ranges.
_____ The target population is all American gun owners.
_____ This is simple random sample because all shooting ranges had equal probability of being selected.
_____ This is a cluster sample because we don't select all shooting ranges.
_____ This is a stratified sample because we sample everyone that is member of the selected shooting ranges.
_____ The sampling frame is the list of all shooting ranges.
g. In all they sampled 189 individuals and found that 123 used guns as a means of home defense. Construct a 98% confidence interval for the proportion of gun owners that use guns for home defense.
h. If we decide to change the confidence level to 95%, what would happen to the size of the margin of error? [1 mark]
_____ It would decrease
_____ It would remain the same
_____ It would increase
_____ Not enough information to tell
i. What concerns, if any, do you have about how the data were collected?
j. The group would now like to find what percentage of the population supports their cause (tougher gun control laws). They would like to obtain a 99% confidence interval with margin of error of 0.03. What sample size should they use to ensure that these demands are met?
6. How popular is the iPhone among college students? During the first year when the iPhone 2G (the first generation) came to the market, 22% of college students owned one. This summer, the second generation, iPhone 3G, was released. A recent survey of 500 college students revealed that 25% owned this latest model.
Based on the survey results, is there sufficient evidence that the iPhone 3G was more popular among college students than the 2G model when each initially appeared on the market? Carry out a hypothesis test on a population proportion at the 5% significance level.
(a) Which of the following is the population proportion of interest? We will denote this population proportion by p. Check only one answer. [1 mark]
____ Whether a college student owns an iPhone 3G or not.
____ The proportion of 500 college students who own an iPhone 3G.
____ The proportion of all college students who own an iPhone 3G.
____ The proportion of all college students who own an iPhone 2G.
(b) Which of the following hypotheses should you test? Check your answer. [2 marks]
_____ Ho : p = 0.25 versus HA : p > 0.25
_____ Ho : p = 0.22 versus HA : p = 0.25
_____ Ho : p = 0.22 versus HA : p > 0.22
_____ Ho : p = 0.25 versus HA : p ≠ 0.25
(c) Find the p-value. Be sure to show all steps.
(d) Decide if the null hypothesis is rejected, and draw a conclusion in the context of the question.
(e) Based on the result of your hypothesis test, which type of error are you in a position of committing? Circle your answer.
i. A Type I error only.
ii. A Type II error only.
iii. Both Type I and Type II errors.
iv. Neither Type I nor Type II error