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 Statistics for Decision Making: Case StudyMutual Funds to writer analysis report
Statistics for Decision Making: Case StudyMutual Funds to writer analysis report Essay Example
 Category:Statistics
 Document type:Assignment
 Level:Undergraduate
 Page:4
 Words:2796
Table of Content
Report of Mutual Funds in Australia
A case Study: Mutual Funds, 2013
Location of Institution
Contents
3Executive summary
4Business Problem
4Statistical Problem
5Analysis
53.1 Descriptive statistics
53.1.1 Descriptive statistics for the Rate of return at based on funds characteristics of category
73.1.2 Descriptive statistics for the rate of return based on funds characteristics of risk
83.1.3 Descriptive statistics for the rate of return based on funds characteristics of objective
83.2 Hypothesis testing
103.2.1 Hypothesis testing that returns on funds with no fees are lower than return on funds with fees
123.2.2 Hypothesis testing that expense ratio of funds with no fees is lower than expense ratio of funds with fees
16Conclusion
16Implications
17References
Executive summary
The report provides an analysis of the returns and return rates of 121 mutual funds. The objective is to provide guidelines to help the clients to select a fund based on different characteristics. Descriptive statistics and hypothesis tests were used to calculate average returns and return rates for the different time periods and make inferences on whether or not certain conditions are true in the whole population. The analysis was important because it helped to compare the different mutual funds and determine which one is better than the others. In conclusion, returns for value objective are higher than returns of funds with growth objective and mutual funds in the small cap category have higher returns. I recommend that clients should purchase mutual funds with value objective because they have higher returns than funds with growth objective. I also recommend that clients should purchase mutual funds in the small cap category because they have higher returns.
Business Problem
There are many mutual funds available in the world today. There is a sample of 121 mutual funds from which one should choose one for their retirement account. Using business statistics help to analyze the different mutual funds and help determine the best fund that one should purchase for their retirement account. This help to minimize any loses that may arise when one purchases a poor fund. This decision is arrived at based on the returns; best and worst of the different funds.
Statistical Problem
The statistical methods for business data are descriptive and inferential statistics. Descriptive statistics generally describe the data using measures of central tendency and dispersion as well as the distribution of the data from which one can determine the distribution of the whole population[ CITATION Hol16 l 1033 ]. The distribution is determined by using graphs which will show whether the data is symmetrical or not. Hypothesis testing is used to test two hypotheses about the population[ CITATION Nag17 p 251309 l 1033 ]. The ttest and ANOVA are of great importance in this case study. The null hypothesis is rejected if the pvalue from the ttest is less than the level of significance[ CITATION Tae14 p 3 l 1033 ]. Hypothesis testing help to determine whether there is a difference between the different mutual funds.
Analysis
3.1 Descriptive statistics
3.1.1 Descriptive statistics for the Rate of return at based on funds characteristics of category
The descriptive statistics for large cap category show that there is a big difference in the average rates of return for the different time periods. The variance is a measure of variability which shows how close the values are to the mean. The variances for the three time periods are 84.2568, 15.9802 and 39.8883. There is higher variability in 2013. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficient of skewness is greater than 0. The distributions are positively skewed.
Return 2013 
3YrReturn 
5YrReturn 

32.77380952 
7.097619048 
1.342857143 

Standard Error 
1.416290076 
0.616832116 
0.974537282 
Standard Deviation 
9.178608735 
3.997528999 
6.315723424 
Sample Variance 
84.2468583 
15.9802381 
39.88836237 
Kurtosis 
0.332418832 
0.313400203 
0.632009343 
Skewness 
0.164818032 
0.572323599 
0.395427196 
The descriptive statistics for mid cap category show that there is a big difference in the average rates of return for the different time periods. The variances for the three time periods are 170.6205, 20.9636 and 80.5888. There is higher variability in 2013. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are not equal to zero. The distributions for 2013 and 5year period are positively skewed while for 3year period is negatively skewed.
Return 2013 
3YrReturn 
5YrReturn 

46.19474 
15.01578947 
6.210526316 

Standard Error 
2.996669 
1.050404076 
2.059493703 
Standard Deviation 
13.06218 
4.578605217 
8.977124926 
Sample Variance 
170.6205 
20.96362573 
80.58877193 
Kurtosis 
0.409484 
0.194173255 
0.729359969 
Skewness 
0.724428 
0.313910951 
0.178490855 
The descriptive statistics for small cap category show that the averages for the 3 and 5year period do not have a huge difference but the average for 2013 is bigger than for the two time periods. The variances for the three time periods are 128.1773, 30.5069 and 85.1431. There is higher variability in 2013. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are not equal to zero. The distribution for 2013 is positively skewed while for 3 and 5year period are negatively skewed.
Return 2013 
3YrReturn 
5YrReturn 

48.38333333 
15.69166667 
12.71666667 

Standard Error 
1.461604966 
0.71305538 
1.191239602 
Standard Deviation 
11.32154339 
5.523303226 
9.227302279 
Sample Variance 
128.1773446 
30.50687853 
85.14310734 
Kurtosis 
0.89564758 
1.002631349 
0.582436593 
Skewness 
0.322850439 
0.393947311 
0.264281608 
3.1.2 Descriptive statistics for the rate of return based on funds characteristics of risk
The descriptive statistics for average risk show that there is a big difference in the average rates of return for the different time periods. The variances for the three time periods are 41.8782, 192.4330 and 65.5101. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are not equal to zero. Since the coefficients are positive, the distributions are positively skewed.
3YrReturn 
Return 2013 
5YrReturn 

42.96304348 
4.247826087 

Standard Error 
0.95414681 
2.045318346 
1.19337033 
Standard Deviation 
6.471338519 
13.87202394 
8.093831372 
Sample Variance 
41.87822222 
192.4330483 
65.51010628 
Kurtosis 
0.383804036 
0.334783597 
0.889110579 
Skewness 
0.301086481 
0.586115957 
0.633148778 
The descriptive statistics for high risk show that there is a big difference in the average rates of return for the different time periods. The variances for the three time periods are 58.0190, 165.2606 and 104.0426. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are not equal to zero. Distributions for 2013 and 3year period are negatively skewed while distribution for the 5year period is positively skewed.
3YrReturn 
Return 2013 
5YrReturn 

12.61764706 
45.99411765 
1.458823529 

Standard Error 
1.84739951 
3.117885673 
2.473893972 
Standard Deviation 
7.617023311 
12.85537196 
10.20012615 
Sample Variance 
58.01904412 
165.2605882 
104.0425735 
Kurtosis 
0.571708967 
0.711316025 
0.397848037 
Skewness 
0.055033838 
0.552536602 
0.781679105 
The descriptive statistics for low risk show that there is a big difference in the average rates of return for the different time periods. The variances for the three time periods are 58.0190, 165.2606 and 104.0426. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are not equal to zero. The distribution for the 3year period is negatively skewed while distributions for the 5year period and 2013 are positively skewed.
3YrReturn 
Return 2013 
5YrReturn 

13.15517241 
41.36206897 
12.36551724 

Standard Error 
0.765084927 
1.631595606 
1.136943695 
Standard Deviation 
5.826713208 
12.42586193 
8.658705217 
Sample Variance 
33.95058681 
154.4020448 
74.97317604 
Kurtosis 
0.688652279 
0.092747083 
0.996457714 
Skewness 
0.152800329 
0.358426997 
0.12986944 
3.1.3 Descriptive statistics for the rate of return based on funds characteristics of objective
The descriptive statistics for growth objective show that there is a big difference in the average rates of return for the different time periods. The variances for the three time periods are 84.2108, 50.7337 and 160.0888. There is higher variability in data for 2013. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are greater than zero. The distributions are positively skewed.
5YrReturn 
3YrReturn 
Return 2013 

2.520408163 
11.09591837 
40.85306122 

Standard Error 
1.310949374 
1.017537373 
1.807517141 
Standard Deviation 
9.17664562 
7.122761613 
12.65261999 
Sample Variance 
84.21082483 
50.73373299 
160.0887925 
Kurtosis 
0.565568342 
0.591673886 
0.414731767 
Skewness 
1.00009196 
0.385748005 
0.321378135 
The descriptive statistics for value objective show that there is a small difference in the average rates of return for the 3 and 5year period. The average for 2013 is higher than for the 3 and 5year period. The variances for the three time periods are 71.3663, 30.5113 and 175.4016. There is higher variability in data for 2013. From the results of skewness, it is evident that the distributions of the data for the three different time periods are not symmetrical since the coefficients of skewness are greater than zero. The distributions 2013 and 5year period are positively skewed. The distribution for 3year period is negatively skewed.
5YrReturn 
3YrReturn 
Return 2013 

11.30416667 
13.62777778 

Standard Error 
0.995589722 
0.650975019 
1.560811842 
Standard Deviation 
8.447858925 
5.523706202 
13.24392765 
Sample Variance 
71.36632042 
30.5113302 
175.4016197 
Kurtosis 
0.847264222 
0.435925684 
0.205049728 
Skewness 
0.111954288 
0.103686289 
0.358645998 
3.2 Hypothesis testing
The steps of hypothesis testing are: setting the hypotheses, deciding on decision rule, calculating the value and then making conclusions[ CITATION Mur14 l 1033 ][ CITATION Met16 l 1033 ].
3.2.1 Hypothesis testing that returns on funds with no fees are lower than return on funds with fees
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average returns on funds with no fees in 2013 and µ_{2 }represents the average returns on funds with fees in 2013
Reject H_{0} if the pvalue is less than 0.05
TTest: TwoSample Assuming Unequal Variances 

With fees 

42.60515 

Variance 
183.5565 
120.597663 
Observations 

0.03131 

P(T<=t) onetail 

t Critical onetail 
At 0.05 level of significance, since the pvalue is 0.4876 > 0.05, fail to reject the null hypothesis and conclude that there is no sufficient evidence to conclude that the returns on funds with no fees are lower than returns on funds with fees is 2013.
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average returns on funds with no fees for 3year period and µ_{2 }represents the average returns on funds with fees for 3year period.
TTest: TwoSample Assuming Unequal Variances 

With fees 

12.41340206 
13.36666667 

Variance 
42.28971435 
30.91884058 
Observations 

0.725958688 

P(T<=t) onetail 

t Critical onetail 

At 0.05 level of significance, since the pvalue is 0.2360 > 0.05, fail to reject the null hypothesis and conclude that there is no sufficient evidence to conclude that the returns on funds with no fees are lower than returns on funds with fees for a 3year period.
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average returns on funds with no fees for 5year period and µ_{2 }represents the average returns on funds with fees for 5year period.
TTest: TwoSample Assuming Unequal Variances 

With fees 

7.59690722 
8.354166667 

Variance 
91.0428028 
113.3730254 
Observations 

0.3182307 

P(T<=t) onetail 

t Critical onetail 
At 0.05 level of significance, since the pvalue is 0.3762 > 0.05, fail to reject the null hypothesis and conclude that there is no sufficient evidence to conclude that the returns on funds with no fees are lower than returns on funds with fees for a 5year period.
3.2.2 Hypothesis testing that expense ratio of funds with no fees is lower than expense ratio of funds with fees
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average expense ratio of funds with no fees and µ_{2 }represents the average expense ratio of funds with fees.
Reject H_{0} if the pvalue is less than 0.05
TTest: TwoSample Assuming Unequal Variances 

With fees 

1.340825 
1.370417 

Variance 
0.221133 
0.112578 
Observations 

0.35444 

P(T<=t) onetail 

t Critical onetail 
At 0.05 level of significance, since the pvalue is 0.3623 > 0.05, fail to reject the null hypothesis and conclude that there is no sufficient evidence to conclude that the average expense ratios of funds with no fees are lower than expense ratios of funds with fees.
3.2.3 Hypothesis testing of whether there is a significant difference between the different categories of the mutual funds
H_{0}: µ_{1}=µ_{2}= µ_{3}
H_{1}:µ_{1}≠µ_{2}≠µ_{3}
Where µ_{1, }µ_{2 }and µ_{3} represent the average returns in 2013 for large, mid and small cap respectively.
Reject H_{0} if the pvalue is less than 0.05
Anova: Single Factor 

Variance 

32.77381 
84.24686 

46.19474 
170.6205 

48.38333 
128.1773 

Source of Variation 

Between Groups 
3153.775 
26.41624 
3.306609E10 

Within Groups 
14087.75 
119.3877 

At 0.05 level of significance, reject the null hypothesis of no difference since the pvalue is less than 0.05 and conclude that there is a significant difference between the average returns across the mutual fund categories in 2013. The mutual fund category which performed better was small cap with an average of 48.3833.
H_{0}: µ_{1}=µ_{2}= µ_{3}
H_{1}:µ_{1}≠µ_{2}≠µ_{3}
Where µ_{1, }µ_{2 }and µ_{3} represent the average returns from 20112013 for large, mid and small cap respectively.
Reject H_{0} if the pvalue is less than 0.05
Anova: Single Factor 

Variance 

7.097619 
15.98024 

15.01579 
20.96363 

15.69167 
30.50688 

Source of Variation 

Between Groups 
1955.988 
977.9942 
40.74342 
3.5164E14 

Within Groups 
2832.441 
24.00374 

4788.429 
At 0.05 level of significance, reject the null hypothesis of no difference since the pvalue is less than 0.05 and conclude that there is a significant difference between the average returns across the mutual fund categories in for the period between 20112013. The mutual fund category which performed better was small cap with an average of 15.6917.
H_{0}: µ_{1}=µ_{2}= µ_{3}
H_{1}:µ_{1}≠µ_{2}≠µ_{3}
Where µ_{1, }µ_{2 }and µ_{3} represent the average returns from 20092013 for large, mid and small cap respectively.
Reject H_{0} if the pvalue is less than 0.05
Anova: Single Factor 

Variance 

1.342857 
39.88836 

6.210526 
80.58877 

12.71667 
85.14311 

Source of Variation 

Between Groups 
3249.257 
1624.629 
23.63981 
2.32327E09 

Within Groups 
8109.464 
68.72427 

11358.72 
At 0.05 level of significance, reject the null hypothesis of no difference since the pvalue is less than 0.05 and conclude that there is a significant difference between the average returns across the mutual fund categories in for the period between 20092013. The mutual fund category which performed better was small cap with an average of 12.7167.
3.2.4 Hypothesis testing that returns on funds with value objective are lower than return on funds with growth objective
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average returns on funds with value objective in 2013 and µ_{2 }represents the average returns on funds growth objective in 2013
Reject H_{0} if the pvalue is less than 0.05
TTest: TwoSample Assuming Unequal Variances 

40.85306122 

Variance 
160.0887925 
175.4016197 
Observations 

1.244453245 

P(T<=t) onetail 
0.108038692 

t Critical onetail 
1.659356034 
At 0.05 level of significance, fail to reject the null hypothesis since the pvalue is greater than 0.05 and conclude that there is no sufficient evidence to conclude that the returns for growth objective are lower than returns for value objective in 2013.
H_{0}: µ_{1}≥µ_{2}
H_{1}:µ_{1}<µ_{2}
Where µ_{1 }represents the average returns on funds with value objective and µ_{2 }represents the average returns on funds growth objective for the period between 2011 and 2013.
Reject H_{0} if the pvalue is less than 0.05
TTest: TwoSample Assuming Unequal Variances 

13.62777778 
11.09591837 

Variance 
30.5113302 
50.73373299 
Observations 

2.09599142 

P(T<=t) onetail 
0.019510916 

t Critical onetail 
1.66276545 
At 0.05 level of significance, reject the null hypothesis since the pvalue is 0.0195 which is less than 0.05 and conclude that there is sufficient evidence to conclude that the returns for growth objective are lower than returns for value objective in 2013.
Conclusion
The analysis based on descriptive statistics and hypothesis tests represent the average rates of return and returns for 2013, 20112013 and 20092013 for the different mutual funds. The average returns for the mutual funds are important to an analyst in determining the best mutual fund that the clients should purchase for their retirement accounts. The mutual fund with the highest returns in a year is the top priority of the any analyst. It is evident that returns for value objective are higher than returns of funds with growth objective and mutual funds in the small cap category have higher returns.
Implications
An analyst should consider the different aspects of a fund before advising the client to reduce losses which may occur from purchasing a mutual fund which performs poorly over the years. Clients should purchase mutual funds with value objective because they have higher returns than funds with growth objective. Clients should purchase mutual funds in the small cap category because they have higher returns.
References
Holcomb, Z.C., 2016. Fundamentals of descriptive statistics. Routledge.
Metler, C.A., & Reinhart, R.V., 2016. Advanced and multivariate statistical methods: Practical application and interpretation. Routledge.
Murphy, K.R., Myors, B., & Wolach, A., 2014. Statistical power analysis: A simple and general model for traditional and modern hypothesis tests. Routledge.
Naghettini, M., 2017. Statistical Hypothesis Testing. In Fundamentals of Statistical Hydrology (pp. 251309). Springer International Publishing.
Taeger, D., & Kuhnt, S., 2014. ‘Statistical hypothesis testing’. Statistical hypothesis testing with SAS and R, pp. 316.