In: Math
Question (statistics) (Data below) (to be done with EVIEWS)
Millions of investors buy mutual funds, choosing from thousands of possibilities. Some funds can be purchased directly from banks or other financial institutions (direct) whereas others must be purchased through brokers (broker), who charge a fee for this service. A group of researchers randomly sampled 50 annual returns from mutual funds that can be acquired directly and 50 from mutual funds that are bought through brokers and recorded their net annual returns (NAR, %), which are the returns on investment after deducting all relevant fees.1 These data are saved in the two columns of the a1.xlsx spreadsheet labelled as Purchase and NAR, respectively. Import these data to EViews.
(a) Are Purchase and NAR qualitative or quantitative variables? If they are qualitative, are they ranked or unranked? If they are quantitative, are they discrete or continuous? What are their levels of measurement? Explain your answers.
(b) Use EViews to obtain the basic descriptive statistics for NAR. Briefly describe what they tell you about the net annual returns from mutual funds.
(c) Using the relevant statistics from part (b), estimate with 90% confidence the mean net annual returns. What assumption do you have to make to perform this task?
(d) Using the relevant statistics from part (b), briefly evaluate whether the assumption needed for the confidence interval in (c) is likely violated.
(e) In general, we can conduct hypothesis tests on a population central location with EViews by performing the (one sample) t-test, the sign test or the Wilcoxon signed ranks test.2 Suppose we would like to know whether there is evidence at the 5% level of significance that the population central location of NAR is larger than 5%. Depending on your answer in part (d), which test(s) offered by EViews would be the most appropriate this time? Explain your answer by considering the conditions required by these tests.
(f) Perform the test you selected in part (e) above with EViews. Do not forget to specify the null and alternative hypotheses, to identify the test statistic, to make a statistical decision based on the p-value, and to draw an appropriate conclusion. If the test relies on normal approximation, also discuss whether this approximation is reasonable this time.
(g) Perform the other tests mentioned in part (e). Again, do not forget to specify the null and alternative hypotheses, to identify the test statistics, to make statistical decisions based on the p-values, and to draw appropriate conclusions. Also, if the tests rely on normal approximation, discuss whether these approximations are reasonable this time.
(h) Compare your answers in parts (f) and (g) to each other. Does it matter in this case whether the population of net returns is normally, or at least symmetrically distributed or not? Explain your answer.
Data
PURCHASE | NAR |
Direct | 9.33 |
Direct | 6.94 |
Direct | 16.17 |
Direct | 16.97 |
Direct | 5.94 |
Direct | 12.61 |
Direct | 3.33 |
Direct | 16.13 |
Direct | 11.20 |
Direct | 1.14 |
Direct | 4.68 |
Direct | 3.09 |
Direct | 7.26 |
Direct | 2.05 |
Direct | 13.07 |
Direct | 0.59 |
Direct | 13.57 |
Direct | 0.35 |
Direct | 2.69 |
Direct | 18.45 |
Direct | 4.23 |
Direct | 10.28 |
Direct | 7.10 |
Direct | 3.09 |
Direct | 5.60 |
Direct | 5.27 |
Direct | 8.09 |
Direct | 15.05 |
Direct | 13.21 |
Direct | 1.72 |
Direct | 14.69 |
Direct | 2.97 |
Direct | 10.37 |
Direct | 0.63 |
Direct | 0.15 |
Direct | 0.27 |
Direct | 4.59 |
Direct | 6.38 |
Direct | 0.24 |
Direct | 10.32 |
Direct | 10.29 |
Direct | 4.39 |
Direct | 2.06 |
Direct | 7.66 |
Direct | 10.83 |
Direct | 14.48 |
Direct | 4.80 |
Direct | 13.12 |
Direct | 6.54 |
Direct | 1.06 |
Broker | 3.24 |
Broker | 6.76 |
Broker | 12.80 |
Broker | 11.10 |
Broker | 2.73 |
Broker | 0.13 |
Broker | 18.22 |
Broker | 0.80 |
Broker | 5.75 |
Broker | 2.59 |
Broker | 3.71 |
Broker | 13.15 |
Broker | 11.05 |
Broker | 3.12 |
Broker | 8.94 |
Broker | 2.74 |
Broker | 4.07 |
Broker | 5.60 |
Broker | 0.85 |
Broker | 0.28 |
Broker | 16.40 |
Broker | 6.39 |
Broker | 1.90 |
Broker | 9.49 |
Broker | 6.70 |
Broker | 0.19 |
Broker | 12.39 |
Broker | 6.54 |
Broker | 10.92 |
Broker | 2.15 |
Broker | 4.36 |
Broker | 11.07 |
Broker | 9.24 |
Broker | 2.67 |
Broker | 8.97 |
Broker | 1.87 |
Broker | 1.53 |
Broker | 5.23 |
Broker | 6.87 |
Broker | 1.69 |
Broker | 9.43 |
Broker | 8.31 |
Broker | 3.99 |
Broker | 4.44 |
Broker | 8.63 |
Broker | 7.06 |
Broker | 1.57 |
Broker | 8.44 |
Broker | 5.72 |
Broker | 6.95 |
(a) PURCHASE is qualitative variables and NAR is quantitative variables because PURCHASE has 2 values viz. "Direct" and "Broker" which are not numerical. It can also be said as a categorical variable.
No, they are not ranked (or ordered).
NAR is a continuous variable as it takes a value within a certain interval.
(b)
Here we can see that the mean is 6.738 and median is 6.16. So, in a crude sense, we conclude that the distribution is more or less positively skewed. And its standard deviation is 4.833074. It implies that on an average all the observations are dispersed by 4.833074 units from the mean value.
(c) Here we conduct Student's one-sample t-test.
Therefore, a 90% confidence limit is (5.935321, 7.540279 )
(d) Here we conduct a normality test (known as Shapiro-Wilk test) to make sure that our normality assumptions hold true or not.
Here the p-value is less than 0.05 so we reject the null hypothesis at 5% level of significance and conclude on the basis of the given data that normality assumption is violated.