Is there a relationship between the number of sleepinghours before the exam and the exam grade? Introduction:  Many ofthe students those days and especially exam days stress themselves on studyingand end up studying all night and getting minimal hours of sleep and verylittle rest. Also, many of them think that it is the right to obtain maximalresults on the exam. Well many studies have shown that it is essential to getenough hours of sleep in order to perform well on the exam and get an optimalgrade. “Dr.

PhilipAlapat, medical director, Harris Health Sleep Disorders Center, and assistantprofessor, Baylor College of Medicine, recommends students instead studythroughout the semester, set up study sessions in the evening (the optimal timeof alertness and concentration) and get at least 8 hours of sleep the nightbefore exams. “Memory recall and ability to maintain concentration are muchimproved when an individual is rested,” he says. “By preparing early and beingable to better recall what you have studied, your ability to perform well onexams is increased.”1However, there has been people that Ipersonally know which does not fit in this fact as they get good grades whilestudying all night long with minimal sleeping hours. This caused me to searchfurther to get extra evidence to know exactly how people are doing. To carryout this task, 50 students , males and females in grade 11 and 12 will be askedto note the numbers of hours they get before an exam and their grade.

Statement of task:The main goal of this task is explore therelationship between number of hours slept before an exam and their performanceon the exam which is determined by the grade. This will show whether there is apositive, negative or no relationship between the two variables. Plan of investigation: The data should be collected and placed in atable in which I plan to use those statistical methods which are Determiningthe statistical basics of central tendencies by calculating the mean, mode,median, lower quartile, upper quartile and interquartile as well as thestandard deviation from the collected results. Also, the graph of the line ofregression.The methods will also include the degree of freedom and correlationco-efficient. Those methods will be done to show whetherthere is a relationship between hours of sleep and the exam grade.

 Data collection: The number of hours slept The grade on exam (%) The number of hours slept   The grade on exam (%) 5 87 6 94 1 66 2 67 9 89 3 75 7 94 4 66 3 77 1 70 9 93 1 75 0 77 1 67 3 69 0 69 2 73 8 86 11 89 9 93 8 96 10 92 7 90 12 75 8 94 12 91 6 93 8 90 6 89 2 66 0 76 4 66 3 68 0 76 5 66 9 98 6 72 6 94 7 88 3 76 9 95 3 70 7 80 9 87 2 66 4 66 2 69 7 87 1 67 3 73         Mathematical processing: Table for midpoint : Number of hours slept Frequency Total Frequency (f1) Midpoint (x1) (f1) * (x1) 0?x<2 9 1 9 2?x<4 12 3 36 4?x<6 5 5 25 6?x<8 10 7 70 8?x<10 10 9 90 10?x<13 4 11.5 46 Total 50   276   Pie chart:         Cumulative frequency table of the hours of sleep:  Hours (hr) Frequency (f) Cumulative frequency 1 9 9 3 12 21 5 5 26 7 10 36 9 10 46 11.5 4 50   Cumulative frequency graph: Mean: It is the sum of all the values divided by thenumber of values. In this case the mean is estimated.Mean: 276/50= 5.

52Mode: it is the most common value in the set of dataand in the table it is the interval with the greatest frequency. The mode is 2?x<4.Median: it is the value that lies in the middle whenthe data are arranged in size order.

Median: (50+1)/2 = 25.5 so the median 5Lower quartile: it isthe median of the lower half of the data.Lower quartile: (50+1)/4 = 12.75 so the lowerquartile 3Upper quartile: it isthe median of the upper half of the data. Upper quartile: 3*(50+1) /4 = 38.25 so theupper quartile is 9The interquartile range: it isthe difference between the lower and upper quartile.IQR=9-3=6  Grade on the exam Frequency Total Frequency (f1) Midpoint (x1) (f1) * (x1) 65

5 1080 70

5 925 95

5 2 50    Mean: It is the sum of all the values divided by thenumber of values. In this case the mean is estimated.Mean: 3970/50= 79.4Mode: it is the most common value in the set of dataand in the table it is the interval with the greatest frequency. The mode is 65

5 so the median 77.5Lower quartile: it isthe median of the lower half of the data.Lower quartile: (50+1)/4 = 12.75 so the lowerquartile 67.5Upper quartile: it isthe median of the upper half of the data. Upper quartile: 3*(50+1) /4 = 38.25 so theupper quartile is 92.5The interquartile range: it isthe difference between the lower and upper quartile.

IQR=92.5-67.5= 25      Standard deviation:   The number of hours slept (X) the grade on the exam (Y) (x-x?) (y-?) (x-x?)2 (y-?)2 (x-x?)(y-?) 5 87 -0.52 7.6 0.2704 57.

76 15.618304 1 66 -4.52 -13.4 20.4304 179.56 3668.

482624 9 89 3.48 9.6 12.

1104 92.16 1116.094464 7 94 1.48 14.6 2.1904 213.16 466.

905664 3 77 -2.52 -2.4 6.3504 5.76 36.578304 9 93 3.48 13.

6 12.1104 184.96 2239.939584 0 77 -5.52 -2.4 30.4704 5.

76 175.509504 3 69 -2.52 -10.4 6.3504 108.16 686.

859264 2 73 -3.52 -6.4 12.3904 40.96 507.510784 11 89 5.48 9.6 30.

0304 92.16 2767.601664 8 96 2.48 16.6 6.1504 275.

56 1694.804224 7 90 1.48 10.

6 2.1904 112.36 246.113344 8 94 2.48 14.

6 6.1504 213.16 1311.019264 6 93 0.48 13.6 0.2304 184.96 42.

614784 6 89 0.48 9.6 0.2304 92.

16 21.233664 0 76 -5.52 -3.4 30.

4704 11.56 352.237824 3 68 -2.52 -11.4 6.3504 129.96 825.297984 5 66 -0.

52 -13.4 0.2704 179.

56 48.553024 6 72 0.48 -7.4 0.2304 54.

76 12.616704 7 88 1.48 8.6 2.1904 73.96 162.001984 9 95 3.48 15.

6 12.1104 243.36 2947.186944 7 80 1.48 0.

6 2.1904 0.36 0.788544 2 66 -3.52 -13.4 12.3904 179.56 2224.

820224 2 69 -3.52 -10.4 12.3904 108.16 1340.145664 1 67 -4.52 -12.4 20.

4304 153.76 3141.378304 6 94 0.48 14.6 0.2304 213.

16 49.112064 2 67 -3.52 -12.4 12.

3904 153.76 1905.147904 3 75 -2.

52 -4.4 6.3504 19.36 122.943744 4 66 -1.52 -13.4 2.

3104 179.56 414.855424 1 70 -4.52 -9.4 20.4304 88.36 1805.230144 1 75 -4.

52 -4.4 20.4304 19.36 395.

532544 1 67 -4.52 -12.4 20.4304 153.76 3141.

378304 0 69 -5.52 -10.4 30.4704 108.16 3295.678464 8 86 2.48 6.6 6.

1504 43.56 267.911424 9 93 3.48 13.6 12.1104 184.96 2239.939584 10 92 4.

48 12.6 20.0704 158.76 3186.376704 12 75 6.48 -4.

4 41.9904 19.36 812.

934144 12 91 6.48 11.6 41.9904 134.56 5650.

228224 8 90 2.48 10.6 6.1504 112.36 691.058944 2 66 -3.52 -13.

4 12.3904 179.56 2224.820224 4 66 -1.

52 -13.4 2.3104 179.56 414.855424 0 76 -5.

52 -3.4 30.4704 11.56 352.237824 9 98 3.48 18.6 12.1104 345.

96 4189.713984 6 94 0.48 14.6 0.2304 213.16 49.112064 3 76 -2.52 -3.

4 6.3504 11.56 73.410624 3 70 -2.52 -9.4 6.3504 88.

36 561.121344 9 87 3.48 7.6 12.1104 57.

76 699.496704 4 66 -1.52 -13.4 2.

3104 179.56 414.855424 7 87 1.48 7.6 2.1904 57.

76 126.517504 3 73 -2.52 -6.4 6.3504 40.96 260.112384 254 3992 581.

36 5948.4 59396.49376                         The above table shows the standard deviation with greatdetail. The r is the Pearson’s product-momentcorrelation coefficient for two sets of data, x and y, is:   Mean of x = 254/50 5.08 Mean of y = 3992/50 79.

84 Sx 3.41 Sy 11.00 Sxy 319.9 r 0.69  r2= 0.47 Coefficient (r) measure of strength in value:0.

0  –0.25          Very weak0.25-0.50          Weak0.50-0.75           Moderate0.

75-1.0            Strong the above data suggests that r= 0.69 which means that itindicates a moderate correlation between the number of hours of sleeping versustheir performance in the exam.

   The line of regression: Regression line is a method that undergoescalculations to find the linear equation of what best fits the data collectedplotted as a scattered graph. It shows both the linear strength and where thecorrelation is negative or positive and can be used to calculated further data/predictions. General formula: y=mx+bThis formula is used to calculate the line ofregression:(y-?)=sxy/(sx)2 (x-x?)(y-79.84)=319.9/(3.41)2(x-5.

08)y= 27.5x– 59.86 As seen in the plotted scattered diagram theplots are close together however some are spread which suggest the moderatelevel of their closeness. The data are slopping upwards suggesting the positivecorrelation between the 2 variables.                Conclusion:  In conclusion, it is seen from the data above that I havesuccessfully managed to prove my hypothesis behind the fact of how sleepinghours and getting rest can affect the performance and thus the grade on anexam. The data showed that there is a correlation which was done through severalmathematical tools that were used throughout the research.

The research wasfirstly done through the collection of data from 50 different students in grade11 and 12. Also, most of the data was done in Microsoft excel in which thestandard deviation was calculated. The correlation was shown to be a moderatepositive correlation which showed that the more hours a student sleeps, thebetter grades will be obtained in the exam.  Validation:  It can be seen how logical and clear the results were whichconstructed a meaningful conclusion. However, there are many limitations thatappeared to be affecting the results in a certain way. The main limitation isthe number of people which were used in the research which was very small to beable to construct a fully accurate conclusion.

Also, there is anotherlimitation which was how some of the numbers were rounded when using thecalculators and therefore it might cause uncertainties.  1 https://www.harrishealth.org/en/news/pages/sleep-key-doing-well-exams.aspx