Category: Probability

A number is chosen from the set of numbers S = {11, 12, 13, 14, 15, 16, 17, 18, 19}. Find the probability
1. the number is even
2.the number is a prime number.
3.the number is odd and less than 17

1. How are permutations different from combinations?
2. Suppose there are 365 days in a year and we are ignoring leap years. Suppose there are n property owners’ club of which you are not a member. You can only become a member if the registration date of your property matches with that of any of the n owners.
a. What is the probability of this match (in terms of n)? Hint: find the probability of the complement event.
b. What should n be for chances of the match being 50%? Why may this number be different from 365/2?
c. Suppose they refuse you a membership in the club. For you to challenge them, you need to show that none of the n existing owners have their registration on the same day. What is the probability of this happening? You may leave the answer as an expression, instead of a number.
3. An insurance company finds that Mark has a 8% chance of getting into a car accident in the next year. If Mark has any kind of accident then the company guarantees to pay him $10, 000. The company has decided to charge Mark a $200 premium for this one year insurance policy.
a. Let X be the amount of profit or loss from this insurance policy in the next year for the insurance company. Find EX, the expected return for the Insurance company? Should the insurance company charge more or less on its premium?
b. What amount should the insurance company charge Mark in order to guarantee an expected return of $100? [10%]
4. Suppose that, some time in the distant future, the average number of burglaries in New York City in a week is 2.2. Approximate the probability that there will be
a. no burglaries in the next week;
b. at least 2 burglaries in the next week.
5. A NYU student claims that she can distinguish Van Leewen ice cream from Hagen Dazs’s ice cream. There are 60% chance of her claim to be true.
a. What is the probability that she needs to test 8 samples to guess the ice cream correctly for the first time. How many ice creams does she need to test on average to arrive at the first correct guess?
b. What is the probability her 8th correct guess comes with the 10th sample that she tastes?

Explain why using self-reported data instead of measured data is a potential pitfall in data collection. Be sure to include an example.

“Suppose you have a bag containing 5 red marbles, 3 blue marbles, and 2 green marbles. You randomly select two marbles from the bag without replacement. What is the probability that the first marble drawn is red and the second marble drawn is blue?”

Instructions
This Discussion Board requires you to answer four questions. For each question, do not simply provide an answer; make sure you explain how you arrived at that answer. Please see the corresponding grading rubric to see how each question is assessed. This is an opportunity for you to show your knowledge and understanding of the weekly concepts. You are strongly encouraged to respond to your classmates’ posts with positive and meaningful feedback and will receive up to one point extra credit per discussion board if you do so. This is meant to be a place where we can learn from one another in an engaging and supportive way!
Questions
1. ESP: For several years, the U.S. General Social Survey asked subjects, “How often have you felt as though you were felt you were in touch with or connected with someone when they were far away from you?” Of 3887 sampled subjects who had an opinion, 1407 said never and 2480 said at least once.
a. Describe the population of interest.
b. Calculate the descriptive statistics (sample proportions).
c. What is the population parameter we want to draw conclusions about (make an inference about)?
2. Use critical thinking to develop an alternative conclusion. A study shows that the number of reported sexually transmitted diseases was significantly higher for high schools that offered courses in sex education than for high schools that did not. Conclusion: The introduction of sex education courses at the high school level has resulted in increased promiscuity among teens.
3. An engineer is designing a machine to manufacture gloves and she obtains the following sample of hand lengths (mm) of randomly selected adult males based on data gathered:
173 179 207 158 196 195 214 199
a. Define this data set as discrete or continuous.
b. Hand lengths are what type of level of measurement?
c. Compare the mean and median for this data set and if you can draw any conclusions from these values.
4. Explain the difference between stratified and cluster sampling. Why do you think that cluster sampling is frequently used in practice?

Many college students are familiar with the term “Freshman 15” which is referring to the average 15-pound (6.8 Kg) weight gain that students my incur during their first year in college. The reasons students may gain weight will vary. Certainly, a change in eating habits, lack of exercise, long hours of studying and being sedentary, stress and anxiety are just a few examples of what can cause weight gain. For this activity, you will use a dataset to evaluate before and after weights for both males and females by conducting summary statistics, graphing your data, and drawing some overall conclusions based on your analysis.
Instructions
Part I:
Conduct this activity in MyLab by PearsonSummary Statistics:
Select “Content” from the Brightspace Navigation toolbar.
Select “MyLab Statistics” from the “Content” menu.
Click on the “MTH 210 StatCrunch” tab
Click on “Open Link”
Select the “StatCrunch Website”
Under the Data Column > select “Data Sets”
Search for the data set “Freshman_15” and select the data
Analyze your data using the Stat pull-down menu:Stat> Summary Stats> Columns
Click “WT SEPT” to move to the right panel
In “Group by”, select “Sex” from drop down menu
Select “Compute!” At the bottom right.
Copy and paste your results into a word document
Repeat the steps above just selecting for b. “WT APRIL” and copy and paste your results into the same word document. Summary statistics for WT SEPT:
Group by: SEX
SEX
n
Mean
Variance
Std. dev.
Std. err.
Median
Range
Min
Max
Q1
Q3
F
35
58.057143
40.467227
6.3613856
1.0752704
57
28
42
70
54
63
M
32
72.71875
110.85383
10.528715
1.8612314
71
43
54
97
65.5
76
Summary statistics for WT APRIL:
Group by: SEX
SEX
n
Mean
Variance
Std. dev.
Std. err.
Median
Range
Min
Max
Q1
Q3
F
35
59.257143
34.373109
5.8628585
0.99100395
58
22
47
69
56
64
M
32
73.875
118.17742
10.870944
1.9217295
71
50
55
105
68
82
Graphs:
You may choose histograms (1) or boxplots (2) to represent your data. If you choose histograms, you will create 4 graphs: WT SEPT Female, WT SEPT Male, WT APRIL Female and WT APRIL Male.
If you choose boxplots, both genders can be represented in the same display for a total of 2 graphs for both months.
Histograms (4 graphs total)Graph > Histogram
Click “WT SEPT” to move to the right panel (repeat all steps again for “WT APRIL”)
In “Group by”, select “Sex” from drop down menu
Select markers (mean and median)
Select “Compute” At the bottom right.
Copy and paste your graphs into a word document (there will be one for F and one for M as you arrow over in your results)
Boxplots (2 graphs total)Graph>Boxplots
Click “WT SEPT” to move to the right panel (repeat all steps again for “WT APRIL”)
In “Group by”, select “Sex” from drop down menu
Select markers (mean and median)
Hit Compute! At the bottom right.
Part II: Interpretation and Conclusions
Answer the following questions in the submission box below:Use your knowledge from Module 1 to discuss the following: How did the average weights change from September to April for males and for females?
Look at the median. Was there a significant shift between the months for males and females?
Describe the shape of the distributions of the graphs for both genders. Were there any potential outliers causing skewness in the data for either month?
Give your overall conclusions based on your analytical results. In this dataset, does it appear that gaining 15 lbs (6.8 Kg) on average is accurate over this time frame (a statistically significant change)? Explain.

A bag contains 5 red marbles, 8 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is either red or green?
To solve this, you can use the formula for the probability of an event ( P(E) ):
P(E)=Number of favorable outcomesTotal number of possible outcomesP(E)=Total number of possible outcomes

Continue working on your project. You will now test your hypotheses using bootstrapping. You should preform two bootstraps one on the categorical and the quantitative. Be sure to include and edits suggested in your instructor’s review or the peer review. Due Monday at Noon.
Please use the data you originally collected for part 1. You will add these new parts to report part 2, 3, and 4.
1. For this project, you must nd some published or existing data. Possible sources include: almanacs, magazines and journal articles, textbooks, web resources, athletic teams, newspapers, professors with experimental data, campus organizations, electronic data repositories, etc. Your dataset must have at least 25 cases, two categorical variables and two quantitative variables. It is also recommended that you are interested in the material included in the dataset.
2. Use bootstrapping to do the analysis.
(a) Compute the standard error for the quantitative variable that you set up the hypothesis test for in part 4 using bootstrapping. Report a 95% confidence interval and decide if you are able to reject or fail to reject your null hypothesis. Create a histogram for your bootstrap distribution.
(b) Compute the standard error for the categorical variable that you set up the hypothesis test for in part 4 using bootstrapping. Report a 95% confidence interval and decide if you are able to reject or fail to reject your null hypothesis. Create a histogram for your bootstrap distribution.2. Use the techniques of the text to repeat your hypothesis test.
3.Use bootstrapping to do the analysis.
(a) Repeat your hypothesis test on the categorical variable utilizing the appropriate formulas for your situation. Compute 95% con dence interval and compare to results from bootstrapping.
4. Add to your report!
(a) Include all items requested above. Include text and graphics describing the processes you have completed.