Analysis of Ice Thrust, Muzzle Velocities and Coconut Palm Production using Statistics

The article provides statistical analysis for ice thrust dependence on ship speed, muzzle velocities of shells with new gunpowder and coconut palm production at four locations in Caribbean using one-way ANOVA.

RESEARCH METHODS: CRASH COURSE IN STATISTICS
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Section B:
1.
a. Ice thrust was found to be positively depending on ship speed with a high significant
positive correlation. The trend indicated a sharp rise in ice thrust along a positive direction
for increase in ship speed. For low speed of 3 meters/second or less very less ice thrust was
observed. Once the ship speed increased to 4 meters/second and above the ice thrust was
noted to range above 250 thousands Newtons. Maximum thrust was observed for ship speed
of 7 meters/second.
876543210
500
400
300
200
100
Ship_speed (Meters/Second)
Ice_Thrust (In thousand Newtons)
Ice Thrust Dependence on Ship Speed
For K1=35
b. The Pearson’s correlation coefficient for the association of ice thrust and ship speed was
found to be positive and significant ( R=0 . 87 , p <0 . 05) .
The correlation value was in line with the trend of the scatter plot of ice thrust and ship
speed, where the positive correlation was evident from the plot.
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The corresponding null hypothesis assumed was that there was no correlation between ice
thrust and speed of ship. The hypothesis was tested at 5% level of significance and the test
statistic of Pearson’s correlation ( R=0 . 87 , p <0 . 05) signified that there was a significant
correlation different from zero at 55 level of significance.
c. The linear regression model was constructed in MINITAB with Ice thrust as the response
variable and ship speed as the predictor variable. The model was found to be statistically
significant ( F=87 . 49 , p< 0 .05 ) at 5% level of significance. The estimated regression
equation was identified as IceThrust =145. 0+40 .28∗ShipSpeed . Hence, for increase in
ship speed for 1 meter/second the thrust of ice was observed to increase by 40.28 thousand
Nektons. The coefficient of determinant or the R-square implied that ship speed was able
to explain 75.76% variation in thrust of ice. The inclusion of adjusted R-square indicated
that the prediction for the variation in response variable was 74.89% for inclusion of the
predictor variable and the predicted R-square was even smaller than the adjusted R-square.
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d. The null hypothesis for the slope of the regression estimating the population slope was that
there was no linear relation between the response and predictor variable as H 0: ( β =0 ) . The
null hypothesis was tested at 5% level of significance against the two tailed alternate
hypothesis ( HA : β≠0 ) . The t-test statistic was evaluated as t=40 . 28−0
4 . 31 =9. 35 and the p-
vale was calculated as P (|t|>9 .35 ) =0 . 000 at 29 degrees of freedom at 5% level of
significance. The results indicated that the p-value was less than alpha value of 0.05, and
hence the null hypothesis was rejected at 5% level. Therefore, it was inferred that ice thrust
and ship speed had significant positive linear association between them.
e. The residual plot revealed that the residuals were aligned along the fitted distribution line.
The points or residuals in the following plot were well within the vicinity of the fitted line
and the distribution seems to be a good fit for the regression analysis.
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100500-50-100
99
95
90
80
70
60
50
40
30
20
10
5
1
Residual
Percent
Normal Probability Plot
(response is Ice_Thrust)
From the “Versus fit plot” between the fitted value and the residuals it was evident that the
pattern of residuals was independent that of the fitted values. The residuals were found to
be scatter randomly along the zero line establishing the fact that the relationship between
the response and the predicting variable was linear and the variances of the error term were
equal. The first two residuals were found to be stand out values from the rest of the values.
This trend was also observed in the scatter plot of ice thrust and ship speed. These two
could have been probable outliers for low speed of ships.
f. The estimated regression equation was IceThrust =145. 0+ 40 .28∗ShipSpeed . Now for
ship speed = 6.8 meters per second, the Ice thrust was found to be
IceThrust =145+40 . 28∗6 . 8=418 . 904 or 418904 Newton.
5

500400300200100
100
50
0
-50
-100
Fitted Value
Residual
Versus Fits
(response is Ice_Thrust)
2. Muzzle velocities of 50 shells tested with a new gunpowder:
a. For detail information for the muzzle velocities of the shells with the new gun powder, the
descriptive statistics were scrutinized. The average muzzle velocity was evaluated as 3007.9
meters per second with a standard deviation or root mean square variation in muzzle
velocities of shells as 42 meters per second. The standard deviation suggested that the new
gunpowder had variable effect on the velocities of the shell with majority of the cells
producing a velocity of 3007.9 meters per second on an average. The median velocity of the
distribution was at 3010.5 meters per second with Interquartile velocity spread as 71.3
meters per second. The mean and median values were pretty close to each other, signifying
that the distribution was almost normal with slight positive skewness in the distribution. In
25% cases velocities for shells with new gunpowder was observed to be less than 2972.1
meters per second, whereas, top 25% shells were able to produce a muzzle velocity greater
than 3043.4 meters per second. The fitted curve over the histogram was noted to be in line
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with the shape of bell curve or normal curve with a single shell producing muzzle velocity
around 3120 meters per second. The 50% of the shells were identified to produce a muzzle
velocity within the range of 2972.1 m/s and 3043.4 m/s.
312030803040300029602920
10
8
6
4
2
0
Mean 3008
StDev 41.98
N 50
Muzzle_Velocities
Frequency
Histogram (with Normal Curve) of Muzzle_Velocities
309030603030300029702940
9
8
7
6
5
4
3
2
1
0
Mean 3006
StDev 38.94
N 49
Muzzle_Velocities
Frequency
Histogram (with Normal Curve) of Muzzle_Velocities
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b. With 95% confidence it was possible to estimate that the average muzzle velocity of shells
in general would be somewhere between [ x
¿
−t0 . 25∗ s
√ n , x
¿
+t 0 .25∗ s
√ n ] or [ 2996. 0 , 3019 . 9 ]
meters per second. Hence, with 95% confidence it was possible to states that target average
velocity of 3000 m/s was well within the limits of the test data. Therefore, at 5% level of
significance it was not possible to reject the null hypothesis assuming average muzzle
velocity of the shells as 3000 m/s.
c. The null hypothesis was considered as H0: μ=3000 against the two tailed alternate
hypothesis of HA: μ≠3000 at 5% level of significance. The hypothesis was tested by one
sample t-test and the t-statistic was evaluated as ( t=1. 33 , p=0 . 188 ) . There was no statistical
evidence at 5% level of significance that the average muzzle velocity of the shells with a
new variant of gunpowder was different from 3000 meters per second.
The muzzle velocities of the shells were found to be normally distributed and the histogram
plot in part a signified this fact. The velocities were continuous and normally distributed in
nature, establishing the validity of the assumptions of one sample t-test.
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3. Matrix representing the distance (in meters) between eight colonies of tropical plants.
a. From the distance matrix in Minitab:
I. Distance between the Abuta and Maracuza colonies is 12.5 meters.
II. Distance between the Gervão and Brazilian Pepper Tree colonies is 55.3 meters.
III. The two closest colonies were Abuta and Cascarilla with distance of 8.7 meters.
b. Cluster analysis with k-means is a possible multivariate analysis technique which could be
used to find the co-ordinates to produce a map of the locations of all the colonies.
c. Cluster Observation in Minitab was performed and the output has been provided below.
Model development with increase in distance level and decreasing similarity level were
identified in Minitab environment for progressive development of clustering up to that point
where at distance = 62.7 meters all the colonies were included in a single cluster.
Dendograms for all the cluster levels have provided with the Minitab output of cluster
observation.
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73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 62.7
73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 50.7
73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 26
73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 12.5
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73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 11.9
73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 10
73648521
62.70
41.80
20.90
0.00
Observations
Distance
Dendogram of Eight Colonies of Tropical Plants with D = 8.7
d. Dendogram Analysis:
i. At 26 meters Abuta, Cascarilla, Maracuza and Gervão colonies form a single cluster.
ii. At 15 meters we would have 3 clusters, the first would consists of Abuta, Cascarilla,
Maracuza and Gervão, the second cluster would consist of Cedro Rosa and Zanga Tempo,
and the third cluster would consist of Brazilian Pepper Tree and Tiririca.
iii. From the cluster observation it was noted that minimum distance for obtaining exact two
clusters was 50.7 meters of distance. One of the clusters would consist of Brazilian
Pepper-Tree and Tiririca, and the second cluster consisted of rest of the six colonies of
plants.
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4. One-way ANOVA in Minitab corresponding to the yields of the Cocos nucifera.
Introduction
Productions of Cocos nucifera or coconut palm at four locations of Caribbean were
compared. Four locations were Jamaica, Turks & Caicos Islands, Granada and Puerto Rico,
where average productions of coconut palm were compared to identify that place where
coconut palm production was significantly the highest. A comparative analysis was
performed with a one-way ANOVA to identify the place with highest coconut palm
production.
Data Analysis
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The average production at four locations were evaluated as 914.18 kilograms per hectare
(SD = 10.15) at Jamaica, 933.27 kilograms per hectare (SD = 9.25) at Turks and Caicos,
940.30 kilograms per hectare (SD = 8.99) at Granada, and 944.92 kilograms per hectare (SD
= 9.23) at Puerto Rico. To test the equality of variances Levene’s test was performed at 5%
level of significance, and the null hypothesis assuming equal variances between the four
average productions failed to get rejected (L = 0.08, p = 0.971). Confidence intervals for
average productions of all four places were found to overlap each other, and the result of
Leven’s test was established at 5% level (Tintle et al., 2015).
Probability plots for productions at all the four locations were drawn using Minitab. It was
observed that the data points for all the four cases were located within the 95% confidence
interval of the fitted average line for coconut palm production. Therefore, productions at
each of the four locations were noted to follow normal distribution. No outlier production
value was identified from the probability plots.
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The null hypothesis was constructed with the assumption that H0: Average Coconut Palm
production at all the four places was same. It was tested at 5% level of significance against the
two tallied alternate hypothesis: HA: that there was a significant difference in average
productions in at least one of the four places. A one-way ANOVA with four groups yielded that
there was a very strong significant difference (F = 47.86, p < 0.05) in average production of
coconut palms at the four places at 5% level rejecting the null hypothesis. Four response
variables were found explain 58.70% variance in productions of the model.
A pair wise comparison was performed by Tukey’s test to identify the pair wise production
difference. From the confidence interval plots it was easily interpreted that there was no
significant difference in average production between Perto Rice and Granada. But, otherwise,
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average coconut palm production at Puerto Rico was extremely and significantly higher
compared to other two locations.
3020100-10-20-30
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1
0.1
Residual
Percent
Normal Probability Plot
(responses are Jamaica, Turks&Caicos, ...)
Puerto_Rico - Granada
Puerto_Rico - Turks&Caicos
Granada - Turks&Caicos
Puerto_Rico - Jamaica
Granada - Jamaica
Turks&Caicos - Jamaica
403020100
If an interval does not contain zero, the corresponding means are significantly different.
Tukey Simultaneous 95% CIs
Difference of Means for Jamaica, Turks&Caicos, ...
Conclusion
Production levels at Puerto Rico and Granada were significantly higher than other two cities.
But, no significant difference in coconut palm farming was observed between the two places.
These two places were almost growing similar quantity of Cocos nucifera. Production in Jamaica
was found to be the lowest and comparative analysis with Turks and Caicos yielded that Turks
and Caicos was the third location after Puerto Rico and Granada to produce Cocos nucifera. All
the analyses were done with 95% confidence, but the evidences were significant enough at 1%
level of significance also.
Section C: Critical Appraisal
Introduction
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The research paper on “Does childhood motor skill proficiency predict adolescent fitness?”
by Barnett et al. (2008) was selected on the basis of identifying a suitable article with a
subject that is most advantageous and was defined in the statistical analysis. To participate
in an in-depth discussion, the validity and reliability of the research literature of the
technical field were analyzed. To find the article in the journal, we looked for journals with
a clinical and medical background, which point to an increase incompetence because of the
skills acquired in the first phase of life. The scientist identified a very limited study in the
selected area.
Critical Summary
The purpose of this article is a review of physical fitness in all its aspects. In 2000, children's
abilities for battery skills were assessed as part of a primary school intervention. Participants
in 2006/2007 were reviewed as part of the study on physical activity and the condition of
cardiovascular respiration measured with the multistage fitness test. Linear regression was
used to examine the relationship between motor and cardiorespiratory fitness of adolescents
for the two genders. All supplements to the structured training was conducted by the study
coordinator and a trainer. During the test procedure, those demonstrating students' improved
motor skills were demonstrated. A multi-level fitness test was chosen over other field
measurements of cardiac respiratory endurance, which were time-wise and distance-
traveled. The sex had no influence on the ability. The most basic assessment for adolescent
motor skills works for a participant who has not lost to follow-up in the future, which is
better known to us.
The data were analyzed using established conventions of regression analysis. The number of
functions classified as existing or correct is summarized for each subject. Each skill, with
the exception of the sprint, was then standardized to a score of 5, and the scores for the six
skills were added to score 15 scores for the three object controls and three locomotive skills.
The researchers of this article were all qualified for cardiovascular fitness in terms of the
number of rounds completed.
Appraisal of Methodology
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In a longitudinal study, the relationship between the child's motor skills and the adolescent's
subsequent cardiorespiratory fitness was examined. Recent perspectives suggest that basic
motor skills are related to children and adolescents, and there is evidence that teens with
weaker motor skills have lower cardiorespiratory endurance.
The revised document was a much wider study and it was part of a longitudinal cohort
design that could use a valid and reliable measurement of cardiovascular disease to measure
data. Because there was much more data, a more meaningful result could be achieved.
However, only 481 participants used essential information on physical activity and skills
testing. Although only a fraction of the participants have been analyzed in detail because
they come from such a large group, they can be considered as a good presentation of two
extremes. The researcher clearly identifies the method, the choice of the participants and the
purpose of the research, and uses known methods to interpret the acquired knowledge. The
data collection method has been followed to maintain unity and concentration throughout
the region.
The research was difficult because some students probably fell out of school. Therefore, the
methods for obtaining information should be such that the participants are flexible enough to
provide good data. The credibility of the participant is also a problem because in cases
where a person is exposed to cardiovascular fitness the response is dependent on how he
explains his feelings and effects. Unfortunately, there were few signs of prejudice, where
experience should have taken into consideration the tracking frequency of a third party
provider. This was unavoidable because of the long follow-up period and the difficulty of
finding students migrating between regions or schools.
Synthesis of Statistical Findings
The document was highly aligned with previous results and perspectives. Obviously, the
object control skills observed in the primary school predicted the subsequent fitness levels in
the adolescents, while the competition for infantile reasons did not predict the subsequent
fitness levels. It is surprising that the speed race has not predicted the physical state of the
study. In compliance with gender, it has been observed that infant control competence has
been associated with the physical state of adolescent respiration, which represented 25.9%
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of physical variations. Compared to women, men better controlled their ability to control
objects. A general linear model was adapted to investigate the relationship between the
control of basic motor skills and the cardiovascular state of adolescence since the
cardiovascular condition is considered a dependent variable. Interactions between the
significant variables of motor skills and gender were considered to investigate whether the
relationship between motor skills and physical fitness varied between male and female
students. There are some healthy lessons learned from the responses in which the sprint race
was tested in a model with gender and the concept of gender interactions to see if there is a
relationship between child performances. The sprint and the respiratory cardiovascular
condition of the adolescents were different from the gender.
Conclusion
The study explained the unjustified univariate relationships and the final model of estimates
of adjusted parameters for the relationship between advanced performance with no
association between cardiovascular respiratory status and school class. As a result of half of
the test, advanced sprint performance in childhood examined linear regression to predict
cardiovascular fitness in adolescence. The reliability of the survey was given as kappa = 0.6,
with the physical condition measured by the number of laps completed in the multi-stage
fitness test.
Reference:
Barnett, L.M., Van Beurden, E., Morgan, P.J., Brooks, L.O. and Beard, J.R., 2008. Does
childhood motor skill proficiency predict adolescent fitness?. Medicine & Science in Sports &
Exercise, 40(12), pp.2137-2144.
Tintle, N., Chance, B.L., Cobb, G.W., Rossman, A.J., Roy, S., Swanson, T. and VanderStoep, J.,
2015. Introduction to Statistical Investigations: High School Binding. John Wiley.
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