3For years, breakfast has been labeled the “most important meal of the day.” Breakfast companies create enticing commercials, depicting happy families and children eating Cheerios to promote “heart health,” or orange juice workers bringing juice straight from the fields to your fridge. Teachers and parents encourage their children to eat breakfast, insisting that it improves overall health and mental capacities. Their claims are not without cause--Harvard University research has found that children who ate a healthy breakfast had higher energy levels and better learning ability than similar students who did not eat breakfast. Another study conducted by Harvard researchers found that students who ate breakfast were "…significantly more attentive in the classroom, earned higher grades in math, and had significantly fewer behavioral and emotional problems.“ However, how significant is the impact of breakfast on academic performance, if it exists at all? To find out the answer, we developed a survey and statistical analysis, after eating a hearty breakfast of course.
4IntroductionWe studied the relationship between eating breakfast and academic success.To gather data, we surveyed students and asked them whether they consistently eat breakfast, what they eat, and what their GPA is.We classified what they eat as healthy or unhealthy according to set standards (ex. fruit, eggs, milk, and oatmeal were considered healthy and cinnamon rolls and doughnuts were considered unhealthy.We plan to study how eating or not eating breakfast, and the type of food that is eaten, affects GPA.
5BiasDuring our survey we also asked for the respondent’s grade level, in order to insure we fully represented each age group.Our sample size is greater than 30, and therefore the CLT should hold, insuring that our results will be approximately normal.Our survey was given randomly to students; however, voluntary response bias may occur.The surveys were administered as an SRS, and individuals can only answer one, making sure the responses are independent.Non-response bias also occurred.
8Analysis of Those Who Eat Breakfast Consistently Without considering the outliers, the distribution of those who eat breakfast consistently is slightly skewed left. The min is 2.0 and the max is 4.8, with a median of 4.1 and a Q1 of 3.8 and a Q3 of 4.41.
9Modified Boxplot of Those Who Eat Breakfast Consistently When the 3 outliers (2.0, 2.8, 2.8) are excluded, the distribution is roughly normal with a min of 3.8, max of 4.8, median of 4.2, Q1 of 3.9 and Q3 ofModified Boxplot
10Boxplot of Those Who Don’t Eat Breakfast Consistently The boxplot of those who don’t eat breakfast is approximately normal with a min of 2, a max of 4.4, a median of 3.8, a Q1 of 2.7 and a Q3 of There are no outliers.
11Conditions to Use Two-Sample T-Test We have 2 SRSs from distinct populations.The samples are independent of each other.The same variable is measured for both samples.The mean and standard deviations of the populations are unknown.
12CalculationsRunning calculations on a calculator (2 variable stats), we determined the mean GPA of breakfast eaters is and the mean GPA of those who don’t eat breakfast is 3.430Xy= Sy=.716Xn= Sn=.715Our null hypothesis is that the GPA of those who do and don’t eat breakfast will be equal: μy = μnOur alternative is that breakfast eaters will have higher GPAs:μ y > μn
13T= / ((.716^2/19)+ (.715^2/19))^.5T=Df=18Tcdf ( ,10^10,18)=P=We used the conservative approach in establishing our degrees of freedom. A two sample t test on the calculator gave us a p value of
14Conclusion About Eating Breakfast Our statistical conclusion is significant at the 5% alpha level (.02<.05).Therefore it is reasonable to reject the null hypothesis and support the alternative that people who consume breakfast consistently will have a higher GPA than those who do not.Because our sample size is not incredibly large, there may be slight bias and further tests should be run to draw a generalized conclusion.
17Analysis of Those Who Eat Healthy Breakfasts In the boxplot of those who eat healthy breakfasts, without considering the outlier at 2.8, the min is 2.8, the max is 4.8, the median is 4.2, Q1 is 3.9 and Q3 is 4.5. This distribution is slightly skewed left.Excluding the outlier at 2.8, the min is 3.8, the max is 4.8, the median is 4.2, Q1 is 3.9 and Q3 is 4.5. This distribution is slightly skewed right.
18Analysis of Those Who Eat Unhealthy Breakfasts The distribution of those who eat unhealthy breakfasts is slightly skewed left with no outliers. The min is 2, the max is 4, the median is 3.335, Q1 is 2.4 and Q3 is
19Healthy vs. UnhealthyNull Hypothesis: Healthy breakfast eaters and unhealthy breakfasts result in the same GPA (μ h = μ uh)Alternative Hypothesis: Those who eat healthy breakfasts will have higher GPAs than those who do not eat healthy breakfasts (μ h > μ uh)
20Calculations Mean of healthy breakfasts: 4.157 Standard Deviation: .497Mean of unhealthy breakfasts: 3.168Standard Deviation: .946TWO SAMPLE T TEST (see previous equation and conditions that apply)( )/((.497^2/15)+(.946^2/4))^.5t=2.02tcdf (2.02,10^10, 3)p=.0683
21Conclusion About Healthy Breakfasts P=.0683Our data is not significant at the 5% alpha level; however, it is significant at the 10% alpha level.There is good, but not overwhelming, evidence to reject the null in favor of the alternative, stating that healthy breakfasts result in higher GPAs.Our sample size for unhealthy breakfasts was very small and may have resulted in some bias, additional tests with a greater sample size will be more helpful in establishing more solid statistical conclusions.
22Final ConclusionOur data showed that those who eat breakfast do indeed have higher GPAs, and those who eat a healthier breakfast have a slight edge over those who love starting their day with big bowls of sugar.Therefore, it is advantageous to eat any type of breakfast, because it helps you start your day and grades off right.