Nutritional Data Set
Abstract/Introduction:
Fast food restaurants are ideally popular for their flavorful, quicktoserve foods that taste good and are primarily inexpensive. On the contrary, these foods tend to be cheap due to the ingredients being used that contain high fat and sugar content, and refined grains instead of healthy constituents like vegetables and whole grains. As researched from the United States Department of Agriculture (USDA), a doublepatty hamburger is, on average, 354 calories. To investigate this, nutritional data of hamburgers from ten popular fast food restaurants were collected to ultimately determine, “Do fast food chains serve more calories, on average, with their advertised hamburgers than the typical hamburger?” With this, my null hypothesis (H_{O}) is μ=354 and my alternate hypothesis (H_{A}) is μ>354, where μ is the true mean caloric count for hamburgers served at each fast food restaurant.
Analysis:
Based on the boxplot of the ten fast food chains, Whataburger edged out Hardees for serving the hamburgers with the highest caloric count with a median value of 830 calories. Although, Sonic’s SuperSonic with Mayo burger item toppled the competition with a caloric count of 1240, while White Castle’s The Original Slider had the lowest caloric amount with 140 due to its small serving size of 63 grams. Wendy’s turned out to be the restaurant with the largest spread in terms of IQR, but Sonic had the largest range. McDonald’s, arguably the preeminent fastfood chain in the industry, was centered slightly lower than Dairy Queen, but both had a slightly normal distribution throughout. Although Burger King is skewed right, opposite of Carl’s Jr., the middle 50% of both distributions are identically aligned with each other. There were no outliers as observed in the boxplot. As seen by the boxplot and summary statistics, each fast food chain, other than White Castle, had a higher median/mean value of caloric amount in their hamburgers than the average hamburger as revealed by USDA. The summary statistics show that White Castle is the only restaurant with a mean of 244 calories and a standard deviation of 107, which is lower than the average hamburger.
Summary statistics for Calories:
Group by: Fast Food Restaurant

A parametric independent sample ttest will be performed to determine whether the advertised burgers of fast food chains have more calories than the typical hamburger on USDA (354 calories). Conditions necessary for inference were met: the data samples were collected in random and independent of one another, and the sampling distribution was approximately normal with a sample size of 69 as stated by the Central Limit Theorem (n>30). The null hypothesis (H_{O})is μ=354 and the alternate hypothesis (H_{A}) is μ>354, where μ is the true mean caloric count for hamburgers served at each fast food restaurant. The teststatistic turned out to be 7.98 with degreesoffreedom equaling 68 (conservative estimate). Moreover, the sample mean was 620.20 with the standard error being 33.37. With this, the pvalue concluded to be less than 0.0001. The probability of obtaining a result equal to or more extreme than what was actually observed is less than 0.0001 (.01%), when the null hypothesis is true.
One sample T hypothesis test:
μ : Mean of variable H_{0} : μ = 354 H_{A} : μ > 354 Hypothesis test results:

Conclusion:
Since the pvalue obtained was less than 0.0001, I would reject the null hypothesis (H_{O}) based on a 0.05 significance level, being that the pvalue was less than 0.05 (0.00001<pvalue<0.0001). There is sufficient evidence to conclude that the true mean caloric count for hamburgers served at each fast food restaurant is greater than the average hamburger as researched by USDA. Possible issues that may have affected the validity of the experiment include sampling variability and measurement errors. The data collected did not represent every fast food chain that serves hamburgers in the world, meaning that there is a tendency for the sample statistic to not match the entire population. In addition, the data collected for each hamburger per fast food restaurant may differ in size and shape due to how they are served each time, resulting in variability. To improve study procedures, nutritional data of hamburger caloric count should be recorded from a larger sample size of fast food chains upwards of 100 restaurants to ensure decreased sampling variability and that each hamburger observed be identically compared with another to guarantee accuracy if the study were to be repeated.
Weight Loss Program Data Set
Abstract/Introduction:
Carrying on from the data collected of ten popular fast food chains, 31 subjects who regularly ate at these restaurants decided to change their lifestyle habits and improve their physical health by participating in an 8week weight loss program. With this program, the individuals ate healthier by basing their diets off fruits, vegetables, lean meats, and whole grains. In addition, the subjects performed daily physical activities such as hiking, swimming, and weight lifting. Data was recorded during the 8week weight loss program, noting down each of the 31 individuals’ weights before and after the program. The question to be answered with this is, “Of the 31 individuals tested, did the 8week weight loss program show a substantial difference in weights before and after the program?” My null hypothesis (H_{O}) is μ_{D}=0 and my alternate hypothesis (H_{A}) is μ_{D}≠0, where μ_{D} is the true average difference between the subjects’ weights before and after the 8week program.
Analysis:
Based on the chart of the individuals’ weights, most of the subjects lost a considerable amount of weight, many of which were in the double digits. However, there were some individuals who actually gained weight during the program which were subjects #5, #16, #20, #22, and #27. The normal quantile quantile plot showed a fairly normal distribution with the correlation equaling 0.916 and the points lining up to an approximately straight line. On the contrary, there were a few points that were a little extreme from the plot which may raise questions on the validity of the study. The sample statistics revealed that with a sample size of 31, the mean weight of the individuals before the program was 196.03 lbs. with a standard deviation of 38.06, while the mean weight of the individuals after the program was 189.06 lbs. with a standard deviation of 35.93.
A match paired T hypothesis test along with a paired T confidence interval will provide a indication of how scattered two sets of measurements are. With this, one can conclude whether there are one or two distributions. The statistical paired ttest procedures gives a better simulation as the data from before and after the weight loss program can be presented in pairs. The data was then analyzed as a twotailed ttest, at the 0.05 significance level. Conditions necessary for inference were met: the data samples were collected in random and independent of one another, and the sampling distribution was approximately normal shown by the normal quantile quantile plot with a sample size of 31 as stated by the Central Limit Theorem (n>30). With a 95% confidence interval for the difference in weights before and after the program, μ_{D}= μ_{1} μ_{2} where the mean of the difference is between weight before and weight after. The 95% C.I. concluded that with a degreeoffreedom equaling 30 (conservative estimate), the interval is 6.97 (mean) ± t_{0.025} x 1.57 (standard error), resulting in a final interval of (3.77,10.16). We can be 95% confident that μ, the true population average in weights before and after the program, lies between 3.77 lbs. and 10.16 lbs. Since zero is not a plausible value of the population mean difference, there is strong evidence to conclude that the weight loss program helped the individuals lose weight during those 8 weeks.
Paired T confidence interval:
μ_{D} = μ_{1}  μ_{2} : Mean of the difference between Weight Before and Weight After 95% confidence interval results:

To more formally conclude this result, a matched pair T hypothesis test would reveal how dispersed the measured weight values are. My null hypothesis (H_{O}) is μ_{D}=0 and my alternate hypothesis (H_{A}) is μ_{D}≠0, where μ_{D} is the true average difference between the subjects’ weights before and after the 8week program. The test statistic turned out to be 4.45 with degreesoffreedom equaling 30 (conservative estimate). Furthermore, the sample mean was 6.97 with the standard error being 1.57. With this, the pvalue concluded to be 0.0001. The probability of obtaining a result equal to or more extreme than what was actually observed is 0.0001 (.01%), when the null hypothesis is true.
Paired T hypothesis test:
μ_{D} = μ_{1}  μ_{2} : Mean of the difference between Weight Before and Weight After H_{0} : μ_{D} = 0 H_{A} : μ_{D} ≠ 0 Sample statistics:
Hypothesis test results:

Conclusion:
Since the pvalue obtained was 0.0001, I would reject the null hypothesis (H_{O}) based on a 0.05 significance level, being that the pvalue was less than 0.05 (0.00001<pvalue<0.0005). There is sufficient evidence to conclude that the true average difference between the subjects’ weights before and after the 8week program does not equal zero. Relating back to the 95% confidence interval, both statistical procedures determined that the 8week weight loss program showed a significant difference between the 31 individuals’ weights before and after the study. Possible issues that may have affected the validity of the experiment include measurement errors and biases such as volunteer bias/social desirability bias. The data collected may have varied by whether the subjects observed were compensated for their time participating in the study and whether the subjects tended to state their weights to what is socially acceptable. To improve study procedures, each individual should be determined to perform the 8week program with no extrinsic rewards and should be able to track their diets/workouts daily.
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