Introduction

The impact of smoking on health outcomes, particularly blood pressure, remains a topic of scientific inquiry. In exploring the relationship between smoking behavior and the likelihood of high blood pressure among the residents of Hofn, it is pertinent to clarify the population parameter of interest. The population parameter of interest here would be the difference in the likelihood of high blood pressure between smokers and non-smokers among the residents of Hofn. This parameter aims to quantify how smoking behavior influences the risk of developing high blood pressure in this specific population. The motivation for this study is grounded in a review of existing literature, specifically an study titled “Smoking and blood pressure: A complex relationship”. This piece of literature provides a comprehensive overview of how smoking adversely affects cardiovascular health through various biological mechanisms. It explains that smoking accelerates arterial aging, disrupts lipid metabolism, and alters the autonomic nervous system, all of which can lead to increased blood pressure. Based on information outlined in the literature and what is already known about smoking, it was predicted before looking at any data that smoking is linked with higher blood pressure. Based on the hypothesis formed prior to data analysis, the initial expectation parameter value is higher for smokers compared to non-smokers. This means that among the residents of Hofn, those who smoke are hypothesized to have a higher likelihood of developing high blood pressure than those who do not smoke.

Data Collection Methods

In an observational study aimed at assessing health-related behaviors and conditions among the residents of Hofn, each individual resident served as an observational unit. The primary variables investigated were whether an individual was a smoker and whether they had high blood pressure, both of which are binary categorical variables. Smoker status was determined by participants’ biography written in their profile, while high blood pressure was assigned task to take a blood pressure test. To ensure a representative sample of the town’s population, the study employed a random sampling technique, utilizing a random number generator to select 100 individuals from an up-to-date registry of residents. This approach aimed to give each resident an equal chance of selection, thereby minimizing sampling bias. However, the study faced potential sources of error. Sampling errors could still occur if the town registry was not fully current, leading to an over- or under-representation of certain groups within the population. Also, there might be other factor such as age, and food that can affect blood pressure. Non-sampling errors were also a concern, particularly with regards to the precision of blood pressure measurements because everyone was tested in different time. The response rate was notably high, which contributed positively to the validity of the findings. Each of the 100 randomly selected participants was approached once, and nearly all were willing to provide the necessary information in this initial visit. By collecting all the necessary data in one visit, the study remained straightforward and reliable.

Descriptive Statistics

In the observational study of the residents of Hofn, I focused on two categorical variables: smoking status (whether an individual is a smoker or not) and high blood pressure status (whether an individual has high blood pressure or not).

Data collected in Hofn are summarized in this two-way table:

High Blood Pressure Ideal Blood pressure Total
Smoker 39 20 59
Non-Smoker 12 29 41
Total 51 49 100

The results, summarized in a two-way table, indicated that out of 59 smokers, 39 had high blood pressure, equating to approximately 66.1%. In contrast, only 12 out of 41 non-smokers reported high blood pressure, representing about 29.3%. This stark difference suggested a significant association between smoking and high blood pressure.

SmokeBlood <- rbind(
do(39) * data.frame(BloodPressure = "High", Smoke = "Smoke"),
do(20) * data.frame(BloodPressure = "Ideal", Smoke = "Smoke"),
do(29) * data.frame(BloodPressure = "Ideal", Smoke = "Nosmoke"),
do(12) * data.frame(BloodPressure = "High", Smoke = "Nosmoke")
)

SmokeBlood<-SmokeBlood %>%
  mutate(proportion=.row/ n()) %>% # creating a new variable called proportion
  select(BloodPressure,Smoke,proportion)

head(SmokeBlood,n=3)
#Side-by-side bar plot(stacked), Mosaic plot
bargraph(~Smoke, groups = BloodPressure, data = SmokeBlood, auto.key = TRUE, stack = TRUE)

mosaicplot(Smoke~BloodPressure, data = SmokeBlood )

This visual representation side-by-side bar plot and mosaic plot also indicates association between smoking behavior and the likelihood of having high blood pressure in the population of Hofn.

Analysis of Results

The population of interest in this study consists of the residents of Hofn. This population includes both smokers and non-smokers. The key parameter being investigated is the difference in blood pressure levels between these two groups.

Null Hypothesis is there is no association between smoking behavior and the likelihood of having high blood pressure among residents of Hofn.

\[H_0: \pi_{smoke}-\pi_{nosmoke}=0 \] Alternative Hypothesis is there is an association between smoking behavior and the likelihood of having high blood pressure among residents of Hofn.

\[H_A: \pi_{smoke}-\pi_{nosmoke} \neq0 \] In this setting, a Type I error happens when the study wrongly concludes there is a link between smoking and high blood pressure when there isn’t one. This mistake could lead to unnecessary actions focusing on stopping smoking to manage blood pressure, even if smoking isn’t a real cause of high blood pressure. A Type II error happens when the study misses a real connection between smoking and high blood pressure. This means the study could wrongly suggest that smoking doesn’t affect blood pressure when it actually does.

My measurements can reasonably be considered a representative sample from the population of interest becasue of the use of a random number generator to select participants from an up-to-date registry of Hofn’s residents helps ensure that every individual had an equal chance of being selected. The study involved 100 residents, which is not big but not small which could reasonably cover Hofn. Given the random sampling technique used and the reasonable sample size, the measurements from this study can likely be considered a representative sample of Hofn.

#sample sizes
n.1<-59
n.2<- 51
# counts
success.1<-39
success.2<-12
#sample proportions
p.hat.1 <- success.1/n.1
p.hat.2 <- success.2/n.2
p.hat.diff<-p.hat.1-p.hat.2
p.hat.pooled<- (success.1+success.2)/(n.1 + n.2)
SE.diff.null<-sqrt((p.hat.pooled*(1-p.hat.pooled))/n.1
+ (p.hat.pooled*(1-p.hat.pooled))/n.2)
z <- p.hat.diff/SE.diff.null
cat("Standardized statistic z is",round(z,2))
## Standardized statistic z is 4.47

Standardized statistic z is 4.47. A z-value of 4.47 indicates that the difference between the two proportions is 4.47 standard deviations away from the hypothesized mean difference of zero (assuming no difference). This is a strong indicator of a significant association between smoking and the presence of high blood pressure. In the study on smoking and high blood pressure in Hofn, I used a random number generator to pick participants from the town’s registry. This method helps ensure that our sample fairly represents the overall population, reducing the chance of bias in our findings. I also made sure that there were at least 10 observations for each category in the two-way table, which helps make our results more reliable. Additionally, I looked at a dotplot—a simple graph that shows how data points are distributed. This graph showed that our data were spread out in a way that is typical for normal distribution, which is good because it means we can use certain statistical methods that assume this pattern in the data.

set.seed(1234)
SmokeBlood.null <- do(1000) * diffprop(shuffle(Smoke) ~ BloodPressure,
data = SmokeBlood)
head(SmokeBlood.null, 2)
dotPlot(~ diffprop, data = SmokeBlood.null, groups = (diffprop <= -0.24 | diffprop >= 0.24), width = 0.0001, cex = 2)

right.tailed.p.value<-pnorm(p.hat.diff, 0, SE.diff.null, lower.tail = FALSE)
cat("The right-tailed p-value using the distribution of sample statistic is",right.tailed.p.value)
## The right-tailed p-value using the distribution of sample statistic is 4.003087e-06

The p-value of 4.003087e-06 (0.000004) indicates the probability of observing a difference in the proportion of high blood pressure between smokers and non-smokers as large as or larger than the one seen in the study, assuming that there is no actual association between smoking and high blood pressure.

Given the p-value of 4.003087e-06 (0.000004), which is substantially smaller than the commonly used significance level of 0.05, the statistical decision would be to reject the null hypothesis. The null hypothesis in this context states that there is no association between smoking behavior and the likelihood of having high blood pressure among the residents of Hofn. The extremely low p-value indicates strong evidence against the null hypothesis, leading to the conclusion that there is a statistically significant association between smoking and high blood pressure in this population.

Given the standardized z-statistic of 4.47 and a corresponding right-tailed p-value of 4.003087e-06, I can conclude that there is a statistically significant association between smoking behavior and the likelihood of having high blood pressure among the residents of Hofn. The results strongly suggest that smoking increases the risk of high blood pressure in this population.

# Standard error of the sampling distribution based on individual proportions
SE.diff.CI<-sqrt((p.hat.1*(1-p.hat.1))/n.1+ (p.hat.2*(1-p.hat.2))/n.2) # margin of error for 95% CI
MoE <- 1.96 * SE.diff.CI
LB<-p.hat.diff - MoE # lower limit of 95% CI
UB<-p.hat.diff + MoE # upper limit of 95% CI 
round(cbind(LB,UB),3)
##         LB    UB
## [1,] 0.258 0.593

The confidence interval for the difference in proportions of high blood pressure between smokers and non-smokers in Hofn is [0.258, 0.593].

The confidence interval indicates that we can be 95% certain that the true difference in the proportion of high blood pressure between smokers and non-smokers lies between 0.26 (26%) and 0.59 (59%). This means that smokers are between 26% and 59% more likely to have high blood pressure compared to non-smokers. The fact that zero is not included in the confidence interval is crucial. It signifies that there is a statistically significant difference between the two groups. This interpretation aligns seamlessly with the conclusions drawn in part 5d of my analysis. There, the significance of the association was established using a z-statistic of 4.47 and a p-value of approximately 0.000004, both suggesting a strong statistical link between smoking and an increased likelihood of high blood pressure. The confidence interval complements this by quantifying the extent of the difference between smokers and non-smokers, providing a range of values that likely contains the true difference in the proportions.

Conclusion

In my study conducted in Hofn, I investigated the relationship between smoking and high blood pressure, uncovering significant insights into how these factors are interrelated. The findings clearly demonstrated a substantial association between smoking and an increased likelihood of having high blood pressure. This was evidenced by the confidence interval for the difference in high blood pressure proportions between smokers and non-smokers, which ranged from 26% to 59% and notably did not include zero. This provided strong evidence that smokers are significantly more likely to suffer from high blood pressure compared to non-smokers. Furthermore, the standardized z-statistic of 4.47 and an exceptionally low p-value of 0.000004 also strengthen the evidence of this association. The data aligned well with existing health literature that links smoking with various cardiovascular risks, including high blood pressure, confirming my expectations and bolstering the validity of my conclusions. Given the random sampling approach and a representative sample size, my findings can reasonably be generalized to the entire population of Hofn. However, extending these conclusions to populations with different demographic or population would require careful consideration. Reflecting on the study I conducted, I would consider increasing the sample size in future studies to potentially enhance the findings, especially if the research scope were broadened to include a more diverse population. Additionally, examining factors such as the duration and intensity of smoking might yield deeper insights into their specific impacts on blood pressure. Looking ahead, further research I’m curious about is blood pressure response after drinking alcohol.

Bibliography

Madika, Anne-Laure, and Claire Mounier-Vehier. “Smoking and blood pressure: A complex relationship.” La Presse médicale (1983), vol. 46, no. 7-8 Pt 1, 2017, pp. 697–702, https://doi.org/10.1016/j.lpm.2017.07.001.