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# Statistics for Decision Making: Case Study-Mutual Funds to writer analysis report Essay Example

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Table of Content

Report of Mutual Funds in Australia

A case Study: Mutual Funds, 2013

Location of Institution

## Contents

3Executive summary

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.

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 251-309 l 1033 ]. The t-test and ANOVA are of great importance in this case study. The null hypothesis is rejected if the p-value from the t-test 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.

## 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 3-Yr-Return 5Yr-Return 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 5-year period are positively skewed while for 3-year period is negatively skewed.

 Return 2013 3-Yr-Return 5Yr-Return 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 5-year 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 5-year period are negatively skewed.

 Return 2013 3-Yr-Return 5Yr-Return 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.

 3-Yr-Return Return 2013 5Yr-Return 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 3-year period are negatively skewed while distribution for the 5-year period is positively skewed.

 3-Yr-Return Return 2013 5Yr-Return 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 3-year period is negatively skewed while distributions for the 5-year period and 2013 are positively skewed.

 3-Yr-Return Return 2013 5Yr-Return 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.

 5Yr-Return 3-Yr-Return 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 5-year period. The average for 2013 is higher than for the 3 and 5-year 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 5-year period are positively skewed. The distribution for 3-year period is negatively skewed.

 5Yr-Return 3-Yr-Return 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

H0: µ1≥µ2

H112

Where µ1 represents the average returns on funds with no fees in 2013 and µ2 represents the average returns on funds with fees in 2013

Reject H0 if the p-value is less than 0.05

 T-Test: Two-Sample Assuming Unequal Variances With fees 42.60515 Variance 183.5565 120.597663 Observations -0.03131 P(T<=t) one-tail t Critical one-tail

At 0.05 level of significance, since the p-value 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.

H0: µ1≥µ2

H112

Where µ1 represents the average returns on funds with no fees for 3-year period and µ2 represents the average returns on funds with fees for 3-year period.

 T-Test: Two-Sample Assuming Unequal Variances With fees 12.41340206 13.36666667 Variance 42.28971435 30.91884058 Observations -0.725958688 P(T<=t) one-tail t Critical one-tail

At 0.05 level of significance, since the p-value 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 3-year period.

H0: µ1≥µ2

H112

Where µ1 represents the average returns on funds with no fees for 5-year period and µ2 represents the average returns on funds with fees for 5-year period.

 T-Test: Two-Sample Assuming Unequal Variances With fees 7.59690722 8.354166667 Variance 91.0428028 113.3730254 Observations -0.3182307 P(T<=t) one-tail t Critical one-tail

At 0.05 level of significance, since the p-value 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 5-year period.

## 3.2.2 Hypothesis testing that expense ratio of funds with no fees is lower than expense ratio of funds with fees

H0: µ1≥µ2

H112

Where µ1 represents the average expense ratio of funds with no fees and µ2 represents the average expense ratio of funds with fees.

Reject H0 if the p-value is less than 0.05

 T-Test: Two-Sample Assuming Unequal Variances With fees 1.340825 1.370417 Variance 0.221133 0.112578 Observations -0.35444 P(T<=t) one-tail t Critical one-tail

At 0.05 level of significance, since the p-value 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

H0: µ12= µ3

H11≠µ2≠µ3

Where µ1, µ2 and µ3 represent the average returns in 2013 for large, mid and small cap respectively.

Reject H0 if the p-value 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 3.306609E-10 Within Groups 14087.75 119.3877

At 0.05 level of significance, reject the null hypothesis of no difference since the p-value 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.

H0: µ12= µ3

H11≠µ2≠µ3

Where µ1, µ2 and µ3 represent the average returns from 2011-2013 for large, mid and small cap respectively.

Reject H0 if the p-value 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 3.5164E-14 Within Groups 2832.441 24.00374 4788.429

At 0.05 level of significance, reject the null hypothesis of no difference since the p-value 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 2011-2013. The mutual fund category which performed better was small cap with an average of 15.6917.

H0: µ12= µ3

H11≠µ2≠µ3

Where µ1, µ2 and µ3 represent the average returns from 2009-2013 for large, mid and small cap respectively.

Reject H0 if the p-value 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 2.32327E-09 Within Groups 8109.464 68.72427 11358.72

At 0.05 level of significance, reject the null hypothesis of no difference since the p-value 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 2009-2013. 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

H0: µ1≥µ2

H112

Where µ1 represents the average returns on funds with value objective in 2013 and µ2 represents the average returns on funds growth objective in 2013

Reject H0 if the p-value is less than 0.05

 T-Test: Two-Sample Assuming Unequal Variances 40.85306122 Variance 160.0887925 175.4016197 Observations -1.244453245 P(T<=t) one-tail 0.108038692 t Critical one-tail -1.659356034

At 0.05 level of significance, fail to reject the null hypothesis since the p-value 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.

H0: µ1≥µ2

H112

Where µ1 represents the average returns on funds with value objective and µ2 represents the average returns on funds growth objective for the period between 2011 and 2013.

Reject H0 if the p-value is less than 0.05

 T-Test: Two-Sample Assuming Unequal Variances 13.62777778 11.09591837 Variance 30.5113302 50.73373299 Observations 2.09599142 P(T<=t) one-tail 0.019510916 t Critical one-tail -1.66276545

At 0.05 level of significance, reject the null hypothesis since the p-value 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, 2011-2013 and 2009-2013 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. 251-309). Springer International Publishing.

Taeger, D., & Kuhnt, S., 2014. ‘Statistical hypothesis testing’. Statistical hypothesis testing with SAS and R, pp. 3-16.