Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to infer that a certain condition holds true for the entire population.

The project focuses on the squat weight of lifters at the Sun Devil Fitness Complex (SDFC). After collecting this data, a group of 10 people will be asked for their opinion on this topic, and the average of their responses will be used to conduct hypothesis testing. Conclusions will be drawn based on the findings.

A personal objective regarding this project is to gain a deeper understanding of the leg strength of a college level gym goer. This will give me insights into where I lie in terms of squat performance compared to others at SDFC and how my own strength measures up relative to the people my age.

I analyzed squat weights collected from 50 college-level gym-goers at the Sun Devil Fitness Complex (SDFC) and compare them to the average estimated squat weight from 10 colleagues.

data <- data.frame(squat_weight = c(135, 145, 90, 95, 205, 135, 165, 225, 185, 185,
                                   145, 135, 95, 115, 65, 135, 185, 225, 315, 45,
                                   205, 215, 135, 115, 95, 95, 185, 135, 155, 165,
                                   185, 135, 135, 95, 45, 185, 225, 135, 135, 155,
                                   225, 315, 155, 135, 95, 185, 135, 185, 225, 135))

The average estimated squat weight taken from my 10 colleagues was 163.5 lbs.

From the whole dataset, this is an example of how entries from my data collection looks like:

##    squat_weight
## 1           135
## 2           145
## 3            90
## 4            95
## 5           205
## 6           135
## 7           165
## 8           225
## 9           185
## 10          185

Parameter of interest:

\[ \mu \]

Null Hypothesis (H₀): The population mean of squat weights for working sets is equal to the average opinion of my 10 colleagues.

\[ H_0: \mu = 163.5 \text{ pounds} \]

Alternative Hypothesis (H₁): The population mean of squat weights for working sets is lesser than the average opinion of my 10 colleagues.

\[ H_1: \mu < 163.5 \text{ pounds} \]

Significance level: \( \alpha = 0.05 \)

Test Statistic Formula:

\[ t_0 = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]

Computations:

\[ t_0 = \frac{154.9 - 163.5}{56.86 / \sqrt{50}} = -1.0694742 \]

\[ p\text{-value} = tcdf(-\infty, -1.0694742, 49) = 0.14504811 \]

Decision Rule:

Reject \( H_0 \) if:
1. \( \bar{x} < \mu_0 - t_{\alpha, n-1} \cdot \frac{s}{\sqrt{n}} \)
2. \( t_0 < -t_{\alpha, n-1} \)
3. \( p\text{-value} < \alpha \)

Since \( p = 0.145 > 0.05 \), we fail to reject the null hypothesis.
There is not enough evidence to claim that the true population mean squat weight is less than 163.5 pounds at the 5% significance level.

The observed sample mean of 154.9 pounds does not provide significant evidence to conclude that the population mean is less than 163.5 pounds at the 5% significance level.

Since the p-value 0.145 > 0.05, we fail to reject the null hypothesis. This means there is not enough evidence to claim that the true population mean squat weight is lower than 163.5 lbs.

t.test(data$squat_weight, mu = 163.5, alternative = "less")

## 
##  One Sample t-test
## 
## data:  data$squat_weight
## t = -1.0695, df = 49, p-value = 0.145
## alternative hypothesis: true mean is less than 163.5
## 95 percent confidence interval:
##      -Inf 168.3817
## sample estimates:
## mean of x 
##     154.9