Statistics 2120: Introduction to Statistical Analysis Homework 8
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Statistics 2120: Introduction to Statistical Analysis
Homework 8
Instructions:
. Respond to each problem below thoroughly, showing all relevant work.
. Use Python for all calculations. Include a screenshot showing relevant code and output for each part using Python.
. Save your completed work as a PDF which you will use on lecture quizzes in the future.
Problems:
1. At the 1976 Pro Bowl, Ray Guy, a punter for the Oakland Raiders, punted a ball that hung mid-air long enough for officials to question whether the ball was filled with helium. The ball was found to be filled with ‘standard’ air, but since then many have tossed around the idea that a helium-filled football would outdistance an air-filledone. Enter science! Some students at the Ohio State University conducted an experiment to test this myth.They used two identical footballs, one filled with standard air and one filled with helium. Each football was kicked 39 times and the two footballs were alternated with each kick. The dataset helium.csv contains the recorded distance of each kick. We’d like to investigate these data in a few ways.
1. Which inference procedure is appropriate to test if the average kicking distance of a football is more than 20 yards, regardless of its contents? Explain.
2. Does the inference procedure identified in part 1. use the standard Normal (z) or t distribution? Explain.
3. What are the appropriate hypotheses for this test?
4. Conduct the appropriate test.
5. Is the conclusion made from the test in part 4 . reliable? Explain.
6. Construct a 95% confidence interval to estimate the average distance for a kick based on these data; again, ignoring the contents of the ball?
7. Does the interval found in part 6. confirm the conclusion made in part 4.?
8. What is the margin of error of the interval found in part 6.?
9. Considering the magnitude of the range of values in the interval found in part 6., does the margin of error found in part 8 . seem large, moderate, or small? Explain.
2. Refer to the football kick distance in Problem 1. A group of kickers and scientists suspect that a ball filled with helium will travel further on average than a ball filled standard air. The file helium.csv also contains the information on the contents of each ball in the fill column. Assign standard air as population 1 and helium as population 2 .
1. Which inference procedure is appropriate to test if a helium filled ball will travel further on average than a ball filled with standard air? Explain.
2. What are the appropriate hypotheses for this test?
3. Conduct the appropriate test.
4. Is the conclusion made from the test in part 3 . reliable? Explain.
5. Construct a 95% confidence interval to estimate the di↵erence in the average kicking distance of standard air and helium-filled footballs.
6. Does the interval found in part 5. confirm the conclusion made in part 3.?
7. What is the margin of error of the interval found in part 5.?
3. Consider a study collecting data from a population with an unknown mean and standard deviation.If the sample mean and sample standard deviation are the same, what is the e↵ect of increasing the sample size on the following measures? The measure can increase, decrease, not change, or more information may be needed.
1. Standard error of the sample mean, SE¯(x)
2. Degrees of freedom.
3. Magnitude of the t statistic.
4. Magnitude of the critical value t* .
5. P-value.
6. Margin of error (assuming the same confidence level) .
7. Width of confidence interval (assuming the same confidence level) .
HW 8
Jessica “Jianan” Xiong (pqf6rd)
On my honor, I did not give nor receive aid on this assignment beyond the listed collaboration.
Problems:
1. At the 1976 Pro Bowl, Ray Guy, a punter for the Oakland Raiders, punted a ball that hung mid- air long enough for officials to question whether the ball was filled with helium. The ball was found to be filled with ‘standard’ air, but since then many have tossed around the idea that a helium-filled football would outdistance an air-filled one. Enter science! Some students at the Ohio State University conducted an experiment to test this myth.They used two identical footballs, one filled with standard air and one filled with helium. Each football was kicked 39 times and the two footballs were alternated with each kick. The dataset helium.csv contains the recorded distance of each kick. We’d like to investigate these data in a few ways.
1. Which inference procedure is appropriate to test if the average kicking distance of a football is more than 20 yards, regardless of its contents? Explain.
A one sample t-test can be used for this study because there is only one quantitative variable: the average recorded distance of footballs. There is also only one population: football.
2. Does the inference procedure identified in part 1. use the standard Normal (z) or t distribution? Explain.
The inference procedure identified in part 1 uses t distribution because the population standard deviation is unknown.
3. What are the appropriate hypotheses for this test?
Null hypothesis: The average kicking distance is less than 20 yards. H0:μ = 20.
Alternative hypothesis: The average kicking distance is greater than 20 yards. Ha: μ > 20.
4. Conduct the appropriate test.
From the code we can notice that the test statistics is about 9.931, and the p-value is about 9. 786 * 10^(−16). When the significance level is 5%, the p-value is less than the significance level. Therefore there is sufficient evidence to reject the null hypothesis. We can conclude that the average kicking distance is more than 20 yards.
5. Is the conclusion made from the test in part 4. reliable? Explain.
The conclusion made from the test in part 4 is reliable because there is a sample size that’s large enough (larger than 30).
6. Construct a 95% confidence interval to estimate the average distance for a kick based on these data; again, ignoring the contents of the ball?
A 95% confidence interval to estimate the average distance for a kick from either of these two balls is (24.92, 27.3877).
7. Does the interval found in part 6. confirm the conclusion made in part 4.?
The interval found in part 6 confirms the conclusion made in part 4 because 20 is not contained in the 95% confidence interval. Therefore, we can reject the null hypothesis and conclude that the average kicking distance is more than 20 yards.
8. What is the margin of error of the interval found in part 6.?
The margin of error of the interval found in part 6 is 1.2339.
9. Considering the magnitude of the range of values in the interval found in part 6., does the margin of error found in part 8. seem large, moderate, or small? Explain.
The margin of error found in part 6 looks small because the margin of error is only about 4.718% of the sample mean.
2. Refer to the football kick distance in Problem 1. A group of kickers and scientists suspect that a ball filled with helium will travel further on average than a ball filled standard air. The file helium.csv also contains the information on the contents of each ball in the fill column. Assign standard air as population 1 and helium as population 2.
1. Which inference procedure is appropriate to test if a helium filled ball will travel further on average than a ball filled with standard air? Explain.
A two-sample t-test is appropriate because there are two independent populations, a helium filled ball and a ball filled with standard air. There is one quantitative variable: the average kicking distance of footballs. And there is also one categorical variable: footballs filled with helium and footballs filled with standard air.
2. What are the appropriate hypotheses for this test?
μ1: the average kicking distance of footballs filled with standard air .
μ2: the average kicking distance of footballs filled with helium.
The null hypothesis is a helium filled ball will travel the same distance on average as a ball filled with standard air. H0: μ2 − μ1 = 0
The alternative hypothesis is a helium filled ball will travel further on average than a ball filled with standard air. Ha: μ2 − μ1 > 0
3. Conduct the appropriate test.
From the code, the test statistics is 0.3703, and the p-value is 0.3566. When significance level is 5%, the p-value is larger than the significance level, which means we fail to reject the null hypothesis and reject the alternative hypothesis. We can conclude that there is insufficient evidence to support that a helium filled ball will travel further on average than a ball filled with standard air.
4. Is the conclusion made from the test in part 3. reliable? Explain.
Yes, the conclusion made from the test in part 3 is reliable because the sample size is large enough (greater than 40).
5. Construct a 95% confidence interval to estimate the difference in the average kicking distance of standard air and helium-filled footballs.
A 95% confidence interval to estimate the difference in the average kicking distance of standard
air and helium-filled footballs is (-2.06, 2.98).
6. Does the interval found in part 5. confirm the conclusion made in part 3.?
Yes, the interval found in part 5. confirms the conclusion made in part 3 because 0 is contained in the 95% confidence interval. Therefore we fail to reject the null hypothesis and reject the alternative hypothesis. The conclusion is that a helium filled ball will not travel further on average than a ball filled with standard air.
7. What is the margin of error of the interval found in part 5.?
The margin of error of the interval found in part 5 is 2.523.
3. Consider a study collecting data from a population with an unknown mean and standard deviation.If the sample mean and sample standard deviation are the same, what is the effect of increasing the sample size on the following measures? The measure can increase, decrease, not change, or more information may be needed.
1. Standard error of the sample mean, SEx-
Decrease. Standard error is s/sqrt(n) . As n increases, the denominator will increase, then the standard error will decrease.
2. Degrees of freedom.
Increase. Degrees of freedom is n- 1. If n increases, degrees of freedom will also increase.
3. Magnitude of the t statistic.
Increase. If n increases, the denominator will decrease, so the t statistic will increase.
4. Magnitude of the critical value t∗ .
Decrease. As the sample size increases, t(k) is less spread. T star will decrease .
5. P-value.
Decrease. As the increase in sample size will lead to the increase in t statistics. P-value is calculated by 1-stats.t.cdf, so P-value will decrease.
6. Margin of error (assuming the same confidence level).
Decrease. When t star and standard error both decrease as the sample size increases, margin of error will also decrease.
7. Width of confidence interval (assuming the same confidence level).
Decrease. As the sample size increases, the confidence interval gets narrower. Therefore the width of the confidence interval decreases.
2023-08-14