Are Older Students Generally Taller?

Ben Gebre-Medhin '98

Chris Palmer '99

Lauren Phillips '99

 

Disclaimer

This study was done in an AP Statistics Course with relatively small sample sizes. The validity of such studies must always be questioned. Please keep this in mind if you use the results of this study. BB&N takes no responsibility for any actions based on these results.

Abstract


It is not unusual for a person to assume that a student's height will increases as he or she go through high school. In order to determine if this assumption is correct, three AP statistics students at Buckingham Browne and Nichols School, ran an ANOVA comparing the heights of 9th 10th, 11th, and 12th graders. The null hypothesis was that there is no difference between the heights of students in grades 9 through 12. (Ho: mu9=mu10=mu11=mu12) The alternative hypothesis was simply that there is a difference. The samples for this test were chosen during the first and second lunch periods at BB&N.

Each observation filled in a individual questionnaire that asked for height, sex, and grade. The total sample size was 171. This sample was then broken down by gender and grade. The counts for the males were: 14 in 9th grade, 28 in 10th grade, 18 in 11th grade, and 21 in 12th grade. The counts for the females were, 17 in 9th grade, 30 in 10th grade, 16 in 11th grade, and 27 in 12th grade.

After entering these observations into the computer an ANOVA was calculated to compare the heights of female in grades 9th through 12th. A second test was also done to compare the heights of males. The p-value for the males came out as p=.000, and for the females the p-value was p=.456. The final conclusion was that the difference between the height of males in 9th, 10th, 11th, and 12th grade is statistically significant, therefore Ho was rejected. It was not determined however, that the difference in the height of females was significant, which meant there was not enough evidence to reject Ho.

Due to sampling bias in favor of students from small private high schools and a relatively small sample size, any extrapolation of these results beyond the BB&N high school is discouraged. Other problems that arose were, determining how to survey the seniors who did not have to attend school because of their Senior Spring Project, figuring out how to divide the work up evenly between the three students, and keeping track of the questionnaires that had already been recorded. It was decided that, should the project be redone, a stronger plan should be decided on first, and the questionnaires should be coded with individual numbers.

The Question

When one thinks about the students in a high school, it is commonly assumed that the seniors are taller than the underclass men. An extension of this generalization may be that older students are generally taller, which would make the mean height of ninth, tenth, eleventh, and twelfth graders be unequal. To eliminate the effects of some confounding the males and females will be separated and two tests will be done on each group. An ANOVA is used to help us decide weather the mean height of high school students differs significantly between grades 9-12.

Sampling

To collect the sample necessary to conduct an ANOVA an anonymous survey designed to collect peoples sex (male, female), grade (9, 10, 11, 12), and height (_____feet_____inches), was created.


 Statistics Survey


      Sex: Male Female


      Grade: 9 10 11 12


      Height: ____feet inches

The data was collected on Thursday and Friday May 15 and 16, 1997. On Thursday students were polled while entering and exiting the cafeteria and while eating during first lunch, the period from 11:40-12:10. First lunch is the time in which most lower class men eat. Second lunch lasts from 12:30-1:00, and consists mainly of upperclassmen. During this time data was collected in the same manner. Twelfth graders are not require to be in school during most school days due to Senior Spring Project. In order to collect data from the 12th grade class, two survey randomly passed out questionnaires during a mandatory morning meeting for seniors from 8:05-8:30 on Friday. To ensure that no one student was represented by more than one data point, the survey was given only to students who said they had not already taken the survey. This resulted in a sample of size of 171 (originally 172; a data entry error rendered one point useless).

 

Data

After the survey was conducted the data was entered into Minitab on three separate computers by three students. As each questionnaire was entered it was marked with an "X" to note that it had been recorded. This system worked well to distinguish used and unused questionnaires. However, simply using an "X" was ineffective in coding each data point with an individual mark, such as a number. Individual marking would have made it possible to recheck for data entry errors.

Assumptions

There are three assumptions for the ANOVA test which are as follows:

The first of these assumptions that we have simple random samples is not completely fulfilled. Our sampling method was not intentionally biased but we did not chose a method that allowed every possible sample to be chosen due to some people being absent or not being in the places were we surveyed. I think that our method was close enough that the lack of completely simple random samples does not affect the results.

The third assumption can be tested by checking if the largest sample standard deviation is no greater than twice the smallest standard deviation. This can be checked against the statistics in the data section. In this situation the ANOVA results are approximately correct.

The second assumption is the most complicated. We have no way to tell if each sample is from a normal population as we do not know the population. We can, however, test for normality by using n-scores. N-scores are calculated by taking each data point and finding in what percentile of the data it is in. The n-scores is then the value of the z-score in the same percentile on the standard normal curve. If the data did come from a normal population and we used a random sample (or close) then a scatter plot of the data points versus the n-scores should be linear. Figures 1-8 are the eight scatter plots of the sample data versus their respective scatter plots (teachers note: figures 1-8 not included)

The Test

The following are the results of the ANOVA tests for males and females as reported by Minitab.

 

 Males  Analysis of Variance:

SOURCE DF SS MS F p
FACTOR 3 154.44 51.48 8.04 0.000
ERROR 77 492.81 6.40
TOTAL 80 647.26

LEVEL     N    MEAN    STDEV
m-9-Hgt  14   67.286   2.673
m-10-Hgt 28   70.743   3.111
m-11-Hgt 18   70.333   1.940
m-12-Hgt 21   71.333   1.932

POOLED STDEV = 2.530

Individual 95% Confidence Intervals for Mean Based on Pooled Standard Deviations:
-+---------+---------+---------+-----
 (-----*------)
                  (----*---)
                 (-----*-----)
                     (-----*----)
-+---------+---------+---------+-----
66.0     68.0      70.0       72.0
 Females

 Analysis of Variance:

SOURCE DF SS MS F p
FACTOR 3 16.03 5.34 0.88 0.456
ERROR 86 523.96 6.09
TOTAL 89 539.99

LEVEL      N      MEAN     STDEV
f-9-Hgt   17    65.265     2.040
f-10-Hgt  30    65.483     2.881
f-11-Hgt  16    66.344     2.737
f-12-Hgt  27    65.130     2.003

POOLED STDEV = 2.468

Individual 95% Confidence Intervals for Mean Based on Pooled Standard Deviation:

----------+---------+---------+------
(-----------*-----------)
   (--------*--------)
          (-----------*------------)
 (--------*---------)
----------+---------+---------+------
                    65.0                 66.0         67.0


Discussion

Although we consider the study fairly accurate, there were many possible sources of error. Sampling contained many of the possible errors in this study. First, we failed to measure each individual height, therefore the recorded values are only estimations. It is very likely that many of the height values are inaccurate. Another problem was that the selection of these students was not completely random. The sample was not an SRS therefore one fundamental assumption was left unfulfilled. Considering the fact that most students were surveyed during the lunch periods in the cafeteria, it is possible that the study did not include students who ate some where else. The second possible source of error came from the data entry. After the data had been collected it was entered into three separate computers. Errors might have been made while recording the heights, grades, and sexes, however we were unable to recheck the data because we did not mark each survey with an individual number. It is also possible that during the manipulation of the data points were lost or changed. Extrapolation is not recommended because many of these errors were very possible


Conclusion

The ANOVA results show that there is significant evidence against Ho for the males, however, Ho for the females can not be rejected. The males had a p-value equal to 0 (p=.000), and the F value was quite large, (F=8.04, df=3), so we can reject Ho at any level. Ho stated that there is no difference between the heights of students in grades 9-12. By rejecting the null hypothesis, we must accept Ha which says that there is a difference between the heights of males in grades 9-12. The F value for the females is small (F=.88, df=3) and the p-value is fairly high, (p=.456).The null hypothesis is not rejected for the females. We concluded that while the study itself was fairly accurate, the results should not be extrapolated, due to the many possible sources of error.

1 The Basic Practice of Statistics. David S. Moore. p. 570.