Statistics 2120: Introduction to Statistical Analysis Homework 6
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Statistics 2120: Introduction to Statistical Analysis
Homework 6
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. In the 2020-2021 season, the UVA men’s basketball team scored an average of 68.6 points per game (SD = approximately 4 points). In the first round of the NCAA men’s basketball tournament, UVA will play (or has played) against Ohio University. Lets suppose that UVA’s head coach has determined that UVA will win if they score at least 70 points.
1. What is the probability that UVA’s next score total (which we will consider as a random sample of the population of all possible score totals) will be at least 70?
2. What is the probability that if UVA played six more games (which is the number of games the winner of the tournament will play), the average score over those six games would be higher than 70 points?
3. What is the probability that the total number of points UVA scores over a random six game stretch is less than 415? Hint: it may be useful to think about this in terms of the average points per game.
2. The file cholesterol.csv contains data from a study that was conducted at a major north eastern American medical center regarding blood cholesterol levels and heart-attack incidents. A total of 37 heart-attack patients had their cholesterol levels measured two days after the attack. The units of cholesterol measurement are not given in the original reference but are presumably mg/dL of blood. Experts in the field do not believe that blood cholesterol levels in patients two days after an attack are Normally distributed, but that they follow a distribution with a non-extreme shape.
1. Assuming that standard deviation of cholesterol levels two days after a heart attack for all patients is 47.7, determine a 99% confidence interval for the mean cholesterol level for all heart attack patients.
2. Does the interval from part 1. provide a reliable estimate for the population mean? Explain.
3. Based on the interval from part 1., is it reasonable to conclude that the average cholesterol level of heart attack patients two days after an attack is greater than 225 mg/dl? Explain.
4. Would your answer to part 3. change if more patients had been included in the study (and the sample mean found was similar)?
5. What is the margin of error of the interval from part 1.?
6. A similar study is being planned at a medical center in the midwest to improve upon the infor- mation gained in the initial study. Researchers would like to limit the resulting 99% confidence interval to a total width of 20 mg/dL. How many heart attack patients should be included in this second study?
3. Refer to the study of blood cholesterol level in heart attack patients two days after their attack described in problem 2.
1. What are the appropriate hypotheses to test if there is evidence that the average cholesterol level of heart attack patients two days after an attack is greater than 225 mg/dL? 2. What are the test statistic and p-value for the hypotheses is part 1.?
3. What is the appropriate conclusion for a 1% significance level? State your conclusion in context. 4. Does your conclusion in part 3. agree with your conclusion in problem 2 part 3.?
HW6
Jessica “Jianan” Xiong
On my honor, I did not give nor receive aid on this assignment beyond the listed collaboration.
Problem 1
1. The probability that UVA’s next score total will be at least 70 is 0.3632.
2. The probability that if UVA played six more games the average score over those six games would be higher than 70 points is 0.1956.
3. The probability that the total number of points UVA scores over a random six game stretch is less than 415 is 0.6357.
Problem 2
1. The 99% confidence interval for the mean cholesterol level for all heart attack patients is (231.6927, 272.0911).
2. The interval from part 1 provides a reliable estimate for the population mean because 99% confidence interval indicates that 99% of the intervals created based on the samples randomly drawn will contain the true population mean. In this case, n is equal to 37 and larger than 30, so sample size is large enough to hold the central limit theorem. The shape of the distribution is approximately normal.
3. It is reasonable to conclude that the average cholesterol level of heart attack patients two days after an attack is greater than 225 mg/dl because from the interval from part 1, 99% of the intervals created will contain the true population mean. The lower bound is 231.6927 mg/dl. Therefore there is a very small chance that the true population mean is lower than 231.6927 mg/dl. And based on this, we can conclude that the average cholesterol level is very likely to be greater than 225 mg/dl.
4. My answer to part 3 will not change if more patients had been included in the study because as the sample size increases, according to the formula, the margin of error will decrease. The confidence interval will be smaller and more accurate, which is located within the previous interval (231.7, 272.1). Therefore we can still conclude that e average cholesterol level of heart attack patients two days after an attack is greater than 225 mg/dl.
5. The margin of error of the interval from part 1 is 20.1992.
6. About 151 heart attack patients should be included in this second study.
Problem 3
1.
Null hypothesis would be the average cholesterol level of heart attack patients two days after an attack is 225 mg/dL. μ = 225
Alternative hypothesis would be the average cholesterol level of heart attack patients two days after an attack is larger than 225 mg/dL. μ > 225
2. The test statistic is 3.4293 and p-value is 0.0003 for the hypotheses.
3. The average cholesterol level of heart attack patients two days after an attack is larger than 225 mg/dL because α = 0. 01 and the p-value 0.0003 which is less than α, the null hypothesis is rejected and the alternative hypothesis is supported.
Based on the data, since p < α , we have sufficient evidence to state that the average cholesterol level of heart attack patients two days after an attack is larger than 225 mg/dL.
4. Yes. Both of the conclusions support that the average cholesterol level of heart attack patients two days after an attack is greater than 225 mg/dl.
2023-08-12