Assignment -

Please answer all multiple-choice and fill-in-the-blank questions.

Q1. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 22.3% and setting the level of significance at 2.5% for the formal hypothesis test. You randomly sample 48 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 1 of the 5-Steps to Hypothesis Testing, choose the appropriate null and alternative hypotheses, i.e. H₀ and H₁, respectively, as well as the significance level, α, pronounced as "alpha".

Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.

a. H₀: p < 0.223, H₀: p = 0.223, α = 1%

b. H₁: p ≠ 0.223, H₀: p = 0.223, α = 1%

c. H₁: p > 0.223, α = 2.5%, H₁: p = 0.223

d. H₀: p = 0.223, H₁: p < 0.223, α = 2.5%

Q2. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 22.2% and setting the level of significance at 10% for the formal hypothesis test. You randomly sample 24 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 2 of the 5-Steps to Hypothesis Testing, choose the appropriate test statistic.

Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.

a. z = (X¯ - µ₀) / (s / √n)

b. z = (X¯₁ - X¯₂) / [S_p * √(1 / n₁ + 1 / n₂) ], where S_p = √( [(n₁ - 1) * s₁² + (n₂ - 1) * s₂²] / [n₁ + n₂ - 2])

c. z = (p^ - p₀) / √(p₀ * (1 - p₀) / n), where p^ = x / n

d. z = (p^₁ - p^₂) / √[p^ * (1 - p^) * (1 / n₁ + 1 / n₂)], where p^₁ = x₁ / n₁, p^₂ = x₂ / n₂, p^ = (x₁ + x₂) / (n₁ + n₂)

Q3. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 1% and setting the level of significance at 5% for the formal hypothesis test. You randomly sample 20 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 3 of the 5-Steps to Hypothesis Testing, choose the appropriate decision rule.

Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.

a. Accept H₀ if z = -2.326

b. Reject H₁ if t < -2.326

c. Reject H₀ if z ≤ -1.645

d. Reject H₀ if t = +1.645

Q4. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 20.6% and setting the level of significance at 5% for the formal hypothesis test. You randomly sample 30 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 4 of the 5-Steps to Hypothesis Testing, compute the test statistic using the appropriate test statistic formula.

Please note the following: 1) 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health; 2) you may copy and paste the data into Excel to facilitate analysis; and 3) do not round your numerical answer that you submit as the online grading system is designed to mark an answer correct if your response is within a given range. In other words, the system does not take into account rounding. On the other hand, rounding is preferable when formally reporting your statistical results to colleagues.

Q5. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 27.3% and setting the level of significance at 10% for the formal hypothesis test. You randomly sample 18 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 5 of the 5-Steps to Hypothesis Testing, choose the appropriate formal and informal conclusions.

Please note the following: 1) 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health; 2) you may copy and paste the data into Excel to facilitate analysis; and 3) in the prior question you already calculated a test statistic but on a different dataset - calculate the test statistic again using the dataset directly above in selecting the corresponding formal and informal conclusions.

a. We accept H₁ because z > +2.326, where z = 1.633. We do not have statistically significant evidence at α = 10% to show that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 27.3%.

b. We accept H₀ because z ≤ -1.282, where z = 1.633. We have statistically significant evidence at α = 1% to show that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 27.3%.

c. We do not accept H₁ because z ≥ -2.326, where z = 1.633. We do not have statistically significant evidence at α = 1% to show that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 27.3%.

d. We do not reject H₀ because z > -1.282, where z = 1.633. We do not have statistically significant evidence at α = 10% to show that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is less than 27.3%.

Q6. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is greater than 6% and setting the level of significance at 2.5% for the formal hypothesis test. You randomly sample 44 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 1 of the 5-Steps to Hypothesis Testing, choose the appropriate null and alternative hypotheses, i.e. H₀ and H₁, respectively, as well as the significance level, α, pronounced as "alpha".

Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.

a. H₁: p = 0.06, H₀: p < 0.06, α = 1%

b. α = 1%, H₀: p = 0.06, H₁: p < 0.06

c. H₀: p ≠ 0.06, H₁: p = 0.06, α = 5%

d. H₀: p = 0.06, H₁: p > 0.06, α = 2.5%

Q7. Previous research states, "no evidence currently exists supporting or refuting the use of electric fans during heat waves" in terms of mortality and illness. Counterintuitively, Public Health guidelines suggest not using fans during hot weather, with some research reporting the potential of fans accelerating body heating.

You decide to research further this seemingly contradictory guidance, hypothesizing that the true population proportion of heart and core temperature increases amidst higher ambient temperature and humidity levels is greater than 42% and setting the level of significance at 5% for the formal hypothesis test. You randomly sample 34 participants based on your research funding and for 45 minutes, the study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. At the end of the 45 minutes, you record for all participants if his/her heart and core temperature increased as compared to the start of the time period. The following table comprises the data you collect.

Per Step 2 of the 5-Steps to Hypothesis Testing, choose the appropriate test statistic.

Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.

a. S_p = √([(n₁ - 1) * s₁² + (n₂ - 1) * s₂²] / [n₁ + n₂ - 2])

b. t = (X¯₁ - X¯₂) / [S_p * √(1 / n₁ + 1 / n₂) ], where S_p = √([(n₁ - 1) * s₁² + (n₂ - 1) * s₂²] / [n₁ + n₂ - 2])

c. t = (X¯ - µ₀) / (s / √n)

d. z = (p^ - p₀) / √(p₀ * (1 - p₀) / n), where p^ = x / n

Q8. Recall in our discussion of the binomial distribution the research study that examined schoolchildren developing nausea and vomiting following holiday parties. The intent of this study was to calculate probabilities corresponding to a specified number of children becoming sick out of a given sample size. Recall also that the probability, i.e. the binomial parameter "p" defined as the probability of "success" for any individual, of a randomly selected schoolchild becoming sick was given.

Suppose you are now in a different reality, in which this binomial probability parameter p is now unknown to you but you are still interested in carrying out the original study described above, though you must first estimate p with a certain level of confidence. Furthermore, you would also like to collect data from adults to examine the difference between the proportion with nausea and vomiting following holiday parties of schoolchildren and adults, which will reflect any possible age differences in becoming sick. You obtain research funding to randomly sample 46 schoolchildren and 48 adults with an inclusion criterion that a given participant must have recently attended a holiday party, and conduct a medical evaluation by a certified pediatrician and general practitioner for the schoolchildren and adults, respectively. After anxiously awaiting your medical colleagues to complete their medical assessments, they email you data contained in the following tables.

What is the estimated 95% confidence interval (CI) of the difference in proportions between schoolchildren and adults developing nausea and vomiting following holiday parties? Assign groups 1 and 2 to be schoolchildren and adults, respectively.

Please note the following: 1) in practice, you as the analyst decide how to assign groups 1 and 2 and subsequently interpret the results appropriately in the context of the data, though for the purposes of this exercise the groups are assigned for you; 2) 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health; 3) you might calculate a CI that is different from any of the multiple choice options listed below due to rounding differences, therefore select the closest match; and 4) you may copy and paste the data into Excel to facilitate analysis.

a. -0.0653 to 0.2519

b. -0.0719 to 0.2771

c. -0.0591 to 0.2834

d. -0.0738 to 0.2368

Q9. Recall in our discussion of the normal distribution the research study that examined the blood vitamin D levels of the entire US population of landscape gardeners. The intent of this large-scale and comprehensive study was to characterize fully this population of landscapers as normally distributed with a corresponding population mean and standard deviation, which were determined from the data collection of the entire population.

Suppose you are now in a different reality in which this study never took place though you are still interested in studying the average vitamin D levels of US landscapers. In other words, the underlying population mean and standard deviation are now unknown to you. You obtain research funding to randomly sample 33 landscapers, collect blood samples, and send these samples to your collaborating lab in order to quantify the amount of vitamin D in the landscapers' blood. After anxiously awaiting your colleagues to complete their lab quantification protocol, they email you the following vitamin D level data as shown in the following table.

What is the estimated 95% confidence interval (CI) of the average blood vitamin D level of US landscapers in ng/mL?

Please note the following: 1) you might calculate a CI that is different from any of the multiple choice options listed below due to rounding differences, therefore select the closest match; 2) ensure you use either the large or small sample CI formula as appropriate; and 3) you may copy and paste the data into Excel to facilitate analysis.

a. 39.1 to 42.4 ng/mL

b. 34.2 to 46.8 ng/mL

c. 33.2 to 38.1 ng/mL

d. 34.8 to 39.8 ng/mL

Q10. Recall in our discussion of the binomial distribution the research study that examined schoolchildren developing nausea and vomiting following holiday parties. The intent of this study was to calculate probabilities corresponding to a specified number of children becoming sick out of a given sample size. Recall also that the probability, i.e. the binomial parameter "p" defined as the probability of "success" for any individual, of a randomly selected schoolchild becoming sick was given.

Suppose you are now in a different reality, in which this binomial probability parameter p is now unknown to you but you are still interested in carrying out the original study described above, though you must first estimate p with a certain level of confidence. You obtain research funding to randomly sample 47 schoolchildren with an inclusion criterion that he/she must have recently attended a holiday party, and conduct a medical evaluation by a certified pediatrician. After anxiously awaiting your pediatrician colleague to complete her medical assessments, she emails you data contained in the following table.

What is the estimated 95% confidence interval (CI) of the proportion of schoolchildren developing nausea and vomiting following holiday parties?

Please note the following: 1) 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health; 2) you might calculate a CI that is different from any of the multiple choice options listed below due to rounding differences, therefore select the closest match; and 3) you may copy and paste the data into Excel to facilitate analysis.

a. 0.180 to 0.509

b. 0.173 to 0.526

c. 0.217 to 0.552

d. 0.205 to 0.476