In: Math
Question (statistics) (Data below) (to be done with EVIEWS or any data processor)
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)
Qualitative is the information about the quality of the data.
Quantitative is the information about the quantity of the data
Nominal - No natural ranking or ordering of the data exists.
Hence, Purchase is a Qualitative variable. They are unranked as no rank is given for the values.
NAR is a Qualitative variable and it is a continuous variable.
Level of Measurement for NAR is interval while for Purchase it is Nominal.
b)
NAR |
|
Mean |
6.74 |
Standard Error |
0.48 |
Median |
6.16 |
Mode |
3.09 |
Standard Deviation |
4.83 |
Sample Variance |
23.36 |
Kurtosis |
-0.59 |
Skewness |
0.55 |
Range |
18.32 |
Minimum |
0.13 |
Maximum |
18.45 |
Sum |
673.78 |
Count |
100 |
From the above summary, Mean of NAV is 6.74 and Std Dev = 4.83
c)
Mean = 4.83
Std Dev = 4.83
Std Error = 0.48
At alpha = 0.1, ZCritical = 1.65
Hence, 90% CI of mean = Mean +/- ZCritical * Std Error = 4.83 +/- 1.65*0.48
= {5.94,7.54}
Assumptions
The dataset is obtained by random selection
The dataset is normally distributed.
d)
Since the sample size is large (greater than 30), hence normal assumption is valid. Also, it is mentioned in the problem that the sample was randomly selected, hence Randomness is also valid.
e)
We should be using the one sample t test as both the necessary assumptions (normality and randomness) are valid in the case.
f)
Null and Alternate Hypothesis
H0: µ = 5
Ha: µ > 5
Onse Sample t Test (Population Std is not known)
X Bar = 6.74
n = 100
s = 4.83
alpha = 0.01
Test Statistic
t = (X Bar - µ)/s/n1/2 = (6.74 –5)/4.83/1001/2 = 3.596
P-value = TDIST(3.596,100-1,1) = 0.000253
Since the p-value is less than 0.05, the data is statistically significant and we reject the null hypothesis ie Population mean is greater than 5.
PS: We are only allowed to answer 4 parts per question.