Statistics 2120: Introduction to Statistical Analysis Homework 3
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Statistics 2120: Introduction to Statistical Analysis
Homework 3
Instructions:
. Be sure to provide your full name and computing ID at the top of your work.
. Write out the Honor Pledge under your name and computing ID: “On my honor, I did not give nor receive aid on this assignment beyond the listed collaboration.”
. List the names of students with whom you collaborated under the Honor Pledge. If you did not collaborate, write ‘None’ .
. Respond to each problem below thoroughly, showing all relevant work.
. Use Python for all calculations. Include a screen shot showing relevant code and output for each part using Python.
. Save your completed work as a PDF and upload it to Gradescope. Be sure to select the appropriate page(s) for each answer. Unselected work will not be graded.
Problems:
1. This Gallup Poll article describes a poll regarding attitudes towards capitalism and socialism among adult Americans. Read the first five paragraphs (until the section “Fewer Than Half of Young Amer- icans View Capitalism Positively”) and the Survey Methods in the grey box at the bottom of the article.
1. What are the response variables?
2. What is the explanatory variable?
3. What is the population of interest for this poll?
4. What is the sample size for this poll?
5. Was this a voluntary response sample?
6. Do you think that there was any undercoverage for this poll?
7. Describe nonresponse in this poll. How would you overcome nonresponse for this poll?
2. Read this summary of the journal article Efect of Two Jumping Programs on Hip Bone Mineral Density in Premenopausal Women.
1. What were the experimental units for this experiment?
2. What were the factor(s), levels, and treatments for this experiment?
3. What is the main response variable?
4. Do you think that the design of this study was biased? Briefly explain.
5. Describe if and how each of the three principles of experimental design were used: . Comparison
. Randomization
. Replication
3. Why are the following statements wrong? Explain.
1. The probability of rain tomorrow is 1.23.
2. Suppose that you checked in your luggage at the airport. The probability of the airline losing your luggage or not losing your luggage is 0.90.
3. Let A and B denote two events. P(A and B) = P(A) ⇥ P(B).
4. Events A and B are disjoint. P(A and B) = 0.2.
4. In this question, a basketball player hits 80% of her free throws. Assume each free throw is indepen- dent of previous and future free throws. Let H denote the event that the player hits a free throw and let M denote the event she misses a free throw.
1. What is the probability of the player missing a free throw, P(M)?
2. Suppose the player attempts two free throws.
i. What is the sample space (or, all the possible outcomes)?
ii. What is the probability that she hits both free throws?
iii. What is the probability that she misses both free throws?
iv. What is the probability that she hits only one free throw out of the pair? 3. Suppose the player attempts three free throws.
i. What is the sample space (or, all the possible outcomes)?
ii. What is the probability that she hits all three free throws?
iii. What is the probability that the first time she misses is on the third attempt? Hint: What does this mean about the first, second, and third attempts?
iv. What is the probability that she would have missed at least one free throw in the three attempts?
Hint: What is the complement?
HW3
Jessica Xiong
‘On my honor, I did not give nor receive aid on this assignment beyond listed collaboration.’ Collaboration: None
Problem 1
1. The response variables are ‘Attitudes towards Socialism’ and ‘Attitudes towards Capitalism’ .
2. The explanatory variable are the political parties people belonged to: Democrats and Republicans.
3. The population of interest in this case is adults who aged 18 and older, living in all 50 U.S. states and the District of Columbia.
4. The sample size for this poll is 1505.
5. This was not a voluntary response sample. This is because voluntary response sample made up of participants who have voluntarily chosen to participate as a part of the sample group. However, in this case, the participants were found using ‘random-digit-dial methods’ .
6. Undercoverage for this poll may be those who are under 18 years old.
7. Nonresponse for this poll can be people who cannot be contacted by ‘random-digit-dial method’ or chooses not to respond and participate in telephone interviews. Ways to overcome this poll may be presenting rewards (little gifts) for people who are willing to participate, thus decreasing the amount of nonresponse.
Problem 2
1. The experimental units for this experiment are sixty premenopausal women, aged 25 to 50 years, completed the intervention.
2.
Factors: Jumping programs
Levels:
1) Jump 10 group : 10 jumps with 30 seconds rest between jumps, twice daily for 16 weeks 2) Jump 20 group : 20 jumps with 30 seconds rest between jumps, twice daily for 16 weeks
3) Control group: did not participate in any jumping exercises and served as the baseline comparison for the other groups.
Treatment: High-impact Jumping Exercises
Treatment 1: Control Group (No specific intervention or jumping program.)
Treatment 2: Jump 10 Group (10 jumps, twice daily, 30 seconds rest between each jump) Treatment 3: Jump 20 Group (20 jumps, twice daily, 30 seconds rest between each jump)
3. The main response variable is hip bone mineral density (BMD) in women.
4. I don’t think this study is biased. This is because this study is described as a randomized controlled trial, which is a strong design to minimize bias and provide a rigorous comparison of interventions.
5.
Comparison:
The principle of comparison was used in this study by comparing the effects of different jumping programs (Jump 10 and Jump 20) with a control group that did not receive any specific intervention. The researchers wanted to determine whether there were significant differences in hip bone mineral density (BMD) between the groups. By having a control group, the researchers could assess the baseline BMD changes and attribute any observed differences to the specific jumping programs rather than other factors.
Randomization:
Randomization was employed in this study. The researchers randomly assigned the premenopausal women, aged 25 to 50 years, to one of the three groups: control, Jump 10, or Jump 20. Random assignment helps ensure that participants have an equal chance of being allocated to any group, which helps balance potential confounding variables and minimize selection bias.
Replication:
Replication was presented in this experiment. It refers to conducting the same experiment with a sufficient number of participants to increase the generalizability and reliability of the results. In this study, the researchers recruited a total of sixty premenopausal women to complete the intervention. The participants were likely distributed across the three groups in a balanced manner, which allows for a fair comparison between the groups. Having a reasonable number of participants helps provide more reliable estimates of the effects of the jumping programs on hip BMD in this population.
Problem 3
The statement "The probability of rain tomorrow is 1.23" is wrong because probabilities are always expressed as values between 0 and 1, inclusive. A probability greater than 1 is not meaningful in the context of probability theory.
The statement "Suppose that you checked in your luggage at the airport. The probability of the airline losing your luggage or not losing your luggage is 0.90" is wrong because the probabilities of all possible outcomes of an event must add up to 1. So, the sum of the probabilities of these two outcomes should be 1. If the probability of the airline losing your luggage is 0.90, then the probability of not losing your luggage should be 1 - 0.90 = 0.10, not 0.90.
The statement "Let A and B denote two events. P(A and B) = P(A) × P(B)" is wrong because it applies to the probability of two independent events, not any two events. The formula P(A and B) = P(A) × P(B) holds true only when events A and B are independent. For dependent events, the correct formula is P(A and B) = P(A) × P(B|A).
The statement "Events A and B are disjoint. P(A and B) = 0.2" is wrong because disjoint events are mutually exclusive, meaning they cannot occur simultaneously. If events A and B are disjoint, then the probability of both events occurring together should be 0.
Problem 4
1. The probability of the player missing a free throw, P(M)=0.2.
2.
i) Sample space (SS) for two free throws: {HH, HM, MH, MM} .
ii) The probability that she hits both free throws is 0.64.
iii) The probability that she misses both free throws is 0.04.
iv) The probability that she hits only one free throw out of the pair is 0.32.
3.
i) Sample space (SS) for three free throws: {HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM}
ii) The probability that she hits all three free throws is 0.512.
iii) The probability that the first time she misses is on the third attempt P(HHM) is 0.128.
iv) The probability that she would have missed at least one free throw in the three attempts is 0.488.
2023-08-10