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1.Quantitative variables can be discrete or continuous. Explain the difference between discrete data and continuous data, and give one example of each.

2.A measure of location is a quantity which is ‘typical’ of the data. Give the names of three such measures, and explain (in words, not formulae) how each is found.

3.What is a measure of spread? Give the names of three such measures.

4.A random sample of a particular attribute yields the histogram shown in figure 1. Suggest a suitable measure of location and a suitable measure of spread for these data.

5.The probability that a ship has a defective radar is 0.05. The probability that a ship has a defective echo is 0.06. Three in one hundred ships have both a defective echo and a defective radar. Find the probability that a randomly chosen ship has either a defective echo or a defective radar.

6.Under what conditions might we use a binomial distribution as a probability model for our data?

7.Under what conditions might we use a normal distribution as a probability model for our data?

8.In hypothesis testing, the p-value can be thought of as the chance of obtaining the observed results, or more extreme results, if the alternative hypothesis is correct. TRUE or FALSE?

9.Write down the assumptions implicit in ANOVA.

10.The scatterplot shown in figure 2 is obtained when observations from two variables are plotted against each other. Choose two words from the following list which might be used to explain the relationship between these two variables:

Correlation analysis of ice thrust and ship speed

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.

The Pearson’s correlation coefficient for the association of ice thrust and ship speed was found to be positive and significant.

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.

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 correlationsignified that there was a significant correlation different from zero at 55 level of significance.

  1. 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 at 5% level of significance. The estimated regression equation was identified as. 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.
  1. 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. The null hypothesis was tested at 5% level of significance against the two tailed alternate hypothesis. The t-test statistic was evaluated as and the p-vale was calculated as 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.
  2. 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.

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.  

  1. The estimated regression equation was. Now for ship speed = 6.8 meters per second, the Ice thrust was found to be or 418904 Newton.
  1. Muzzle velocities of 50 shells tested with a new gunpowder:
  2. 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 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.
  1. With 95% confidence it was possible to estimate that the average muzzle velocity of shells in general would be somewhere between or 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.
  2. The null hypothesis was considered as H0: against the two tailed alternate hypothesis of HA: at 5% level of significance. The hypothesis was tested by one sample t-test and the t-statistic was evaluated as. 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.    

  1. Matrix representing the distance (in meters) between eight colonies of tropical plants.
  1. From the distance matrix in Minitab:
  2. Distance between the Abutaand Maracuza colonies is 12.5 meters.
  3. Distance between the Gervãoand Brazilian Pepper Tree colonies is 55.3 meters.
  • The two closest colonies were Abutaand Cascarilla with distance of 8.7 meters.
  1. 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.
  2. 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. 
  1. Dendogram Analysis:
  2. At 26 meters Abuta, Cascarilla, Maracuzaand Gervão colonies form a single cluster.
  3. At 15 meters we would have 3 clusters, the first would consists of  Abuta, Cascarilla, Maracuzaand Gervão, the second cluster would consist of Cedro Rosa and Zanga Tempo, and the third cluster would consist of Brazilian Pepper Tree and 
  • 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 andTiririca, and the second cluster consisted of rest of the six colonies of plants.
  1. One-way ANOVA in Minitab corresponding to the yields of the Cocos nucifera.

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. 

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). 

Comparison of coconut palm productions among four Caribbean locations

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.  

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, average coconut palm production at Puerto Rico was extremely and significantly higher compared to other two locations.


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.

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.

Probability plots for coconut palm productions

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.

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.

Relationship between motor skill proficiency and adolescent fitness

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.

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% 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.


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. 


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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