STAT670 Introductory Statistics

Introduction

Body fitness is important for both body wellbeing and increasing the body stamina. Researchers have found different factors that affect the muscular strength. One of this factor include gender, age, types of muscle fiber among others (Tackett, 2017). In this research, the main focus will be to determine whether gender, an individual body weight, and age have a significant impact on the weight lifted. To achieve this, the physical and performance data will be used to test the hypothesis. The study will answer the following research questions:

(a) Is there any difference in the average age of male and female weightlifters?

(b) What is the relation between the body weight of weightlifters and the maximum weight they can lift?

All the tests are performed at the 0.05 level of significance.

Methods

Different research analysis method would be incorporated into this research. First, the descriptive statistics would be carried out to determine the measures of central tendency and the measures of dispersion. Second, the assumptions for tests of the hypothesis would be carried out to ensure that most of the crucial ones are met. That is, some of the assumptions such as normality test and equality of variances would be tested. This will help in ensuring that appropriate tests are conducted for valid decisions and conclusion. The first primary test of hypothesis technique that would be applied in the report is independent sample t-test. In which the assessment would be carried out to test whether there is an average difference in the age of the weightlifters by gender. Second, a simple linear regression would be carried out to determine whether the body weight had a significant association with the maximum weight lifted. However, the second test would be carried out after testing the linearity assumption using Pearson’s correlation. Further assessment would be carried out to determine whether all assumptions of running least square model are met.

A sample of 243 weightlifters will be used. Where their gender, age, body weight and maximum weight lifted were recorded. 

Results

The descriptive summary of the variables is as summarized in the table below.

Descriptive Statistics: bodyweight, age, weightlifted

Variable        N  N*    Mean  SE Mean   StDev  Minimum      Q1   Median       Q3  Maximum

bodyweight    243   0   99.040    0.966  15.060   62.000  87.000  100.500  111.700  133.500

age           243   0   27.560    0.325   5.067   12.470  24.100   27.470   31.140   40.300

weightlifted  243   0   146.53     2.03   31.64   104.70  116.30   124.70   178.80   191.00

The results indicate that on average the weightlifters had an average bogy weight of 99.040 kgs (SD = 15.060) (Keller, 2014). The minimum weight was 62kgs and the maximum weight 133.50kgs. The middle 50% of the weight lifters weigh between 87.00kg and 111.70kgs.

On the other hand, the average age of the weightlifters is 27.560 years (SD = 5.067). The age ranges between 12.47 years and 40.30 years. Lastly but not least, on average the maximum weight lifted was 146.53 (SD = 31.64). The minimum weight lifted was 104.70 kilograms and the maximum weight lifted was 191.00 kilograms.

The assessment was carried first to determine whether all the three quantitative data were normally distributed. Their normality plots are as illustrated below.

The results indicate that there is enough evidence to claim that the body weight was not normally distributed (AD = 1.904, p-value < .05). This particularly implies that there was sufficient evidence to reject the null hypothesis which indicates that the data are normally distributed.

The results also pointed that the body weight was not normally distributed (AD = 26.620, p-value < .05). Thus, we can confidently conclude that the weight lighted is none normal.

The normal plot for the age indicates that the age data are normally distributed (AD = 0.260, p-value < .05). This is also evident since most of the data points are within the range and they follow a specific trend.

To carry out the independent t-test, assumption about equality of variance was tested. Note that although the data failed to meet the normality assumption, the t-test is robust about this assumption. Thus, the analysis can still be carried out and valid results obtained (Anderson et al., 2016).  The equality of variance assumption tests results is as follows.

                       Test

Method  DF1  DF2  Statistic  P-Value

Bonett    —    —          —    0.299

Levene    1  241       1.77    0.184

The summary table indicates that the equality of variance between the age of the participants was met (F (1, 241) = 1.77, p-value = 0.184). This means that the male and female weightlifters had equal variance in the age distribution. Thus, the independent t-test will use the assumption of equal variances between the groups.

The independent t-test results are as tabulated below.

Two-Sample T-Test and CI: age, gender

Two-sample T for age

gender    N   Mean  StDev  SE Mean

Female  125  28.13   5.27     0.47

Male    118  26.96   4.79     0.44

Difference = μ (Female) - μ (Male)

The estimate for difference:  1.175

95% CI for difference:  (-0.100, 2.450)

T-Test of difference = 0 (vs ≠): T-Value = 1.82  P-Value = 0.071  DF = 241

Both use Pooled StDev = 5.0435

The results indicate that there is no enough evidence to reject the null hypothesis (T (241) = 1.82, p-value = 0.071). This implies that there is no significant difference in the age between the weightlifters by gender. In particular, the average age between the male and female weightlifters is equal. The 95% confidence interval indicates that the average difference between female and male was between -0.100 and 2.450. Notably, the confidence interval contains a zero. This supports the decision to fail to reject the null hypothesis.

Further, assessment to answer the second research question was carried out. But first, the assumptions were checked. That is, before running simple linear regression analysis was carried out to determine whether there was a linear relationship between the body weight of the weightlifters and the maximum weight lifted. The results are as follows.

Probability Plot of age

Correlation: bodyweight, weightlifted

Pearson correlation of bodyweight and weight lifted = 0.870

P-Value = 0.000

The results depict that there is a strong positive correlation between the body weight and the weight lifted (r = 0.870, p-value < .05). This association is significant. Thus, we can conclude that linearity assumption is met. Hence, a simple linear regression model can be fitted (Chatterjee and Hadi 2015).

The regression model was fitted on the data and the results are as illustrated below.

Regression Analysis: weight lifted versus bodyweight

Analysis of Variance

Source          DF  Adj SS  Adj MS  F-Value  P-Value

Regression       1  183162  183162   747.81    0.000

  bodyweight     1  183162  183162   747.81    0.000

Error          241   59029     245

  Lack-of-Fit  194   52015     268     1.80    0.010

  Pure Error    47    7014     149

Total          242  242191

Model Summary

      S    R-sq  R-sq(adj)  R-sq(pred)

15.6503  75.63%     75.53%      75.28%

Coefficients

Term          Coef  SE Coef  T-Value  P-Value   VIF

Constant    -34.39     6.69    -5.14    0.000

bodyweight  1.8268   0.0668    27.35    0.000  1.00

Regression Equation

weightlifted = -34.39 + 1.8268 bodyweight

Durbin-Watson Statistic

Durbin-Watson Statistic = 1.91624

As indicated earlier, there is a significant relationship between body weight and the weight lifted (F (1, 241) = 747.81, p-value < .05). In other words, there was enough evidence to reject the claim that there is no linear relationship between these two variables. Thus, the body weight is an ideal predictor of the weight lifted. However, further assessment should be carried out on the model, to determine whether it is the most appropriate. First, the model could take into account 75.63% of sources of variation. Based, on this value, this model is reliable to predict weight lifted. That is, the proportion of variation was high enough to warrant the prediction of values which are quite close to the actual values of the dependent variable.

The model indicated that when the weight of the weightlifter increased by one kilogram, he/she was expected to increase the weight lifted by 1.8268 kilograms. The results indicated that those with more weight are expected to light more weight.

Residual diagnostics were carried out and the results indicated that the errors were randomly distributed with no significant trend. In fact, the Durbin-Watson value 1.91624 indicated that the error rate was random. In other words, there was no correlation between the residuals. This is as illustrated below.

The residual plot indicates that the residuals are evenly distributed around the zero. That is, although there seems to be a trend, this relationship is not significant.

Conclusion

The assessment was carried out in alliance with the set research questions. First, although some of the variables, such as the body weight, and weight lifted failed to meet the normality test. The report still obtained meaningful and valid results. First, it was found that there was no evidence that the average age of the weightlifters was significantly different. Thus, it can be concluded that the average age of the weightlifters was the same. Second, the findings pointed that there was a significant relationship between the weightlifter’s weight and the maximum weight lifted. There was a positive relationship, which implied that as the weight of an individual increase, he or she was expected to light heavier weights.

References

Anderson, D.R., Sweeney, D.J., Williams, T.A., Camm, J.D. and Cochran, J.J., 2016. Statistics for business & economics. Nelson Education.

Chatterjee, S. and Hadi, A.S., 2015. Regression analysis by example. John Wiley & Sons.

Keller, G., 2014. Statistics for management and economics. Nelson Education.

Tackett, C. (2017). Factors Affecting Muscular Strength. [online] Afpafitness.com. Available at: https://www.afpafitness.com/research-articles/factors-affecting-muscular-strength [Accessed 29 May 2018].