Math 247 Final Project Report
Ivy Hicks
Introduction
My research question was: Can stretching for 5 minutes improve a persons attention (via an attention test)? The two variables I choose to work with are stretching and attention. I have heard and had many people tell me that exercise can help improve your mood, focus, and even seems to correspond with good grades. Also, many people I know tend to take mini study breaks for 5 minutes or so and go on a little walk or stretch then go back to work. I wanted to study if these small study breaks could actually be associated with an improvement in attention and that is why I choose these variables. The first variable is categorical since it was whether or not a person streched before the attention test. In order to make the attention test into a categorical variable I based it on whether or not a person’s score was the same, less or improved. If someone scored higher the second time they took the test then I considered it improved attention, this was the success case for this variable. Otherwise, if their score was the same or less it was considered a failure and not improved attention. The stretching variable was easy to measure on islanders since I decided whether or not to assign them the task of stretching and holding for 5 minutes. In order to measure their attention I had them complete the 10 minute attention test. In this test they watched a series of letters appear and pressed a button each time they saw the letter Z. The reported number was the number of misses. I had the islanders take this test once to get a baseline result. Then after 60 minutes I had them either stretch and take the test again or just take the test again. I choose 60 minutes because this is how many people I know will split up their studying, they’ll work for 45 to 60 minutes then they take a 5 minute break. I had the control and stretching group start the attention test with the same amount of break so the stretch group took the attention test after they stretched after 55 minutes and then took the test 60 minutes after the first test. The control group took the second attention test 60 minutes after the first. The population parameter of interest was the difference in proportions of improved attention for the stretch group and no stretch group. After looking at literature related to these variables I initially thought that the parameter was going to be higher, the stretching group would have a larger proportion of improved attention than the non stretching group.
The first piece of literature that I read was about linking exercise to better grades. There seemed to be a lot of confounding variables and although there seemed to be an association between exercise and better grades the paper could not draw any causation conclusions (Anon “Exercise linked…”). This paper was a review paper that looked at different studies findings. I found this paper after looking for literature that relates focus and exercise since I am interested in seeing if my friends studying methods could actually cause them to have better attention when they go back to studying after their short break. Another paper I looked at talked about the effects of water based exercise in elderly adults, with a focus on cognitive function. The research found that there was significant improvement in attention in groups that participated in the water activities (Daisuke). This study provided evidence that there may be some link between exercise and attention and made me more confident in posing my research question. Finally, a study I looked at found that yoga and exercise in schizophrenia patients improved cognitive function as well (Anon “A Randomised Controlled…”). This made me feel that stetching would also be a good variable to test since yoga involves a lot of stretching.
Data Collection Methods
The observational units in the study are each of the people. I used a random number generator in order to randomly choose which people in the islands were in my study. First I numbered the large islands 1-3 and generated 40 random numbers between 1 and 3. Then for each of the islands I numbered the towns and did a random number generation to choose a town for the amount of times that large island had been generated. Then I did a random number generation to choose the house in the town. Finally if a household contained more than one person I did random number generation again to choose who would be in the study. If the person did not consent to the study then I rerandomized from the beginning to randomly select a new person to be in the study. I repeated this process until I had 40 randomly selected people. Also, if a house that was generated was empty I would randomize the house generation again. There were two people who were selected that did not have the option to do an attention test. I took them out of the study and randomly selected two new people using my previous method. I then randomly assigned the 40 people to be a part of the stretching or no stretching group, this ended with 20 people in each group. After this first initial round of testing I found that I did not have enough observational units to meet the validity conditions for a theory based test. To remedy this problem I repeated my methods for twenty new observational units to ensure I had at least 10 observations in each box of the two way table.
Once all participants had been assigned groups I started the study. I started the attention test for all participants within seconds of each other. Once the attention test was completed I set a timer for 55 minutes and 60 minutes. During this time I recorded how many misses each person had in their attention test. After 55 minutes I started the 5 minute stretching and holding for all the participants in the stretch group. Once the 60 minute timer went off I once again ran the attention test for all participants. I then recorded the number of misses for each participant and determined if they had improved or not. If someone had the same number of misses this was counted as not improvement. I recorded the final data in long format using y and n to indicate if the participant had improved or not. I also used y and n to indicate if they were part of the stretch group or not.
A potential source of sampling or non-sampling error could have occurred when I was selecting houses and ran into the issue of a house being empty. Instead of randomizing from the beginning I just randomly selected another house in the same town. Also, since I had multiple random number generators I could have made errors when writing down the random numbers that were generated or selecting the correct number house or person.
Descriptive Statistics
| No stretch | Stretch | Total | |
|---|---|---|---|
| No attention improvement | 15 | 19 | 34 |
| Attention improvement | 15 | 11 | 26 |
| Total | 30 | 30 | 60 |
Both of my variables are categorical. The first variable, stretching or no stretching, is categorical since it is whether or not a person stretched before the attention test. In order to make the attention test into a categorical variable I based it on whether or not a person’s score was the same, less or improved. If someone scored higher the second time they took the test then I considered it improved attention, this was the success case for this variable. Otherwise, if their score was the same or less it was considered a failure and not improved attention.
The proportion of those who had improved attention with no stretching was 0.5, this is the value of \(\hat{p}_{no\ stretch}\). The proportion of those who had improved attention scores and stretched was 0.367, this is the value for \(\hat{p}_{stretch}\). The difference in proportions is 0.133. The relative risk of these values is 1.36 by subtracting 1 and multiplying by 100 we get a percent change of 36%. This means that from the stretch to no stretch group there was a 36% increase in proportion of attention improvement. Since there is a moderate percent change and proportion difference between the two groups it seems that there may be some association between the two variables.
StretchAttention <- read_csv("~/Math 247/Final Project/Mini Project 3 Data Collection - Ivy Hicks - Sheet2.csv")
mosaicplot(Improvement ~ Stretch, data = StretchAttention)
Analysis of Results
The population being represented is the entirety of the islands. My measurements can be reasonably considered a representative sample from the population of interest. The population of interest is the entirety of the Islands. I randomly sampled from this entire population by using a random number generator. The parameter of interest is the difference in proportions of those with improved attention in the stretching and no stretching group. The null hypothesis is there is no association between stretching and attention. The alternative hypothesis is that there is an association between stretching and attention. Both hypothesis in symbols are below:
\(H_0: \pi_{stretch} = \pi_{no\ stretch}\)
\(H_A: \pi_{stretch} \neq \pi_{no\ stretch}\)
A type I error would be if the data lead to a conclusion that rejected the null hypothesis when the null hypothesis is actually true. In this study if the data led to the conclusion that there is association between stretching and improved attention but actually there is no association between these two variables, this would be the type I error. A type II error would occur when the null hypothesis is false but the data did not lead to a rejection of the null hypothesis. In this setting if the conclusion reached was to not reject the null hypothesis, that there is no association between stretching and improved attention, but in reality there actually is association between these two variables, this represents a type II error.
I used a theory-based approach for the statistical analysis. The data meets the validity conditions for theory-based approach because there are at least 10 observations in each box of the two way table. The appropriate standardized statistic is the z-statistic because this study is on two binary categorical variables. The z-statistic is 1.039. It was calculated using the following R code:
stretch <- .367
nostretch <- .5
allsuccesses <- (15+11)/60
n1 <- 30
n2 <- 30
z <- (nostretch-stretch)/sqrt(allsuccesses*(1-allsuccesses)*((1/30)+(1/30)))
z## [1] 1.039495I also calculated a two sided p-value which was 0.299. This is the probability that we would observed the same result of this study or a more extreme value if the null hypothesis was true. The R code used to calculate it is below:
pvalue <- pnorm(1.039495, 0, 1, lower.tail = FALSE) + pnorm(-1.039495, 0, 1, lower.tail = TRUE)
pvalue## [1] 0.2985746Since the p-value is greater than 0.1 meaning it provides little evidence against the null hypothesis. It is plausible that there is no association between stretching and improved attention, the null hypothesis is plausible. I also used R to calculate the confidence interval. I am 95% confident that the true difference in proportions between attention improvement in stretching vs not stretching for the islanders population is between -0.148 and 0.415. Since zero is included in this confidence interval there is little to no evidence against the null hypothesis.
confint(prop.test(Improvement ~ Stretch, success="yes", data = StretchAttention))Conclusion
In this study the data did not behave as I expected. I initially thought that the proportion of those with improved attention would be higher in the stretch group but this was not the case. The two sided p-value and confidence interval that I calculated both indicated that there was little/no evidence against the null hypothesis, which was there is no association between improved attention and stretching. My p-value was larger than 0.1 and my confidence interval included 0 which is why the null hypothesis is plausible. This result can be generalized and be used to draw cause and effect conclusions because I used random sampling and random assignment. The population that this can be generalized to is the entirety of the Islands because this is the population I randomly sampled from.
After conducting this study I began thinking about what I would do differently and how I could build upon this study in the future. Something I would do differently would be to collect a larger sample. Ideally I would like this to be a random sample from the entirety of the islands but I also think it could be interesting to conduct this study with random samples from each island and compare the results to see if living on certain islands might be a factor. Some similar questions for further research to build on this study could include the addition of an age constraint for the sample, testing different lengths in time between the attention tests, testing varying exercises and length of exercises.
Bibliography: references to literature mentioned in the introduction
Anon. “A Randomised Controlled Trial of Adjunctive Yoga and Adjunctive Physical Exercise Training for Cognitive Dysfunction in Schizophrenia.” Acta Neuropsychiatrica, vol. 29, no. 2, 2017, pp. 102-114. ProQuest, http://ezproxy.whitman.edu/login?url=https://www.proquest.com/scholarly-journals/randomised-controlled-trial-adjunctive-yoga/docview/2788633374/se-2, doi:https://doi.org/10.1017/neu.2016.42.
“Exercise Linked to Better Grades – the Oakland Post.” University Wire, Jan 31, 2016. ProQuest, http://ezproxy.whitman.edu/login?url=https://www.proquest.com/wire-feeds/exercise-linked-better-grades-oakland-post/docview/2874577458/se-2.
Daisuke, Sato, et al. “Differential Effects of Water-Based Exercise on the Cognitive Function in Independent Elderly Adults.” Aging Clinical and Experimental Research, vol. 27, no. 2, 2015, pp. 149-159. ProQuest, http://ezproxy.whitman.edu/login?url=https://www.proquest.com/scholarly-journals/differential-effects-water-based-exercise-on/docview/2348931956/se-2, doi:https://doi.org/10.1007/s40520-014-0252-9.