Question

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Followings Tables shows previous 11 months stock market returns.

Date	Monthly SP500 Return	Monthly DJIA Return
12/7/2007	-0.8628	-0.7994
1/8/2008	-6.1163	-4.6323
2/8/2008	-3.4761	-3.0352
3/8/2008	-0.5960	-0.0285
4/8/2008	4.7547	4.5441
5/8/2008	1.0674	-1.4182
6/8/2008	-8.5962	-10.1937
7/8/2008	-0.9859	0.2468
8/8/2008	1.2191	1.4548
9/8/2008	-9.2054	-6.0024
10/8/2008	-16.8269	-4.8410

1. Let consider we know the variance of monthly return for all stock were 25 percent in 2008, perform the following hypothesis for each index:

Ho :  µ = -5

Ha :  µ ≠ -5

2. Considering the population variance is unknown, perform a hypothesis test if the average stock return is 0 or not for each index:

Ho :  µ = 0

Ha :   µ ≠  0

3. Perform following hypothesis based on all assumption we had in 2)

Ho :  µ ≥ -3

Ha :   µ <  -3

4. Let’s consider the population mean of SP500 as µ1 and that of DJIA as µ2 while none of population variance is known. Test following hypothesis:

  Ho : µ1  =  µ2

Ha :   µ1 ≠  µ2

5. Let’s consider the population variance of SP500 as σ²₁and that of DJIA as σ²₂, and none of them are known. Test following hypothesis:

  Ho :  σ²₁  =  σ²₂

  Ha :  σ²₁≠  σ²₂

6. Perform the following hypothesis test

  Ho :  σ²₁  ≤  σ²₂

  Ha :  σ²₁>  σ²₂

Answer 1

ans::

as for given data

1) a)

Two-sample T for Monthly vs Monthly DJIA Return

N Mean StDev SE Mean
Monthly 11 -3.60 6.13 1.8
Monthly DJIA Return 11 -2.25 4.05 1.2

Difference = μ (Monthly) - μ (Monthly DJIA Return)
Estimate for difference: -1.36
95% CI for difference: (-5.98, 3.26)
T-Test of difference = -5 (vs ≠): T-Value = 1.64 P-Value = 0.116 DF = 20
Both use Pooled StDev = 5.1956

b)

Two-sample T for Monthly vs Monthly DJIA Return

N Mean StDev SE Mean
Monthly 11 -3.60 6.13 1.8
Monthly DJIA Return 11 -2.25 4.05 1.2

Difference = μ (Monthly) - μ (Monthly DJIA Return)
Estimate for difference: -1.36
95% CI for difference: (-5.98, 3.26)
T-Test of difference = 0 (vs ≠): T-Value = -0.61 P-Value = 0.547 DF = 20
Both use Pooled StDev = 5.1956

c)

Two-sample T for Monthly vs Monthly DJIA Return

N Mean StDev SE Mean
Monthly 11 -3.60 6.13 1.8
Monthly DJIA Return 11 -2.25 4.05 1.2

Difference = μ (Monthly) - μ (Monthly DJIA Return)
Estimate for difference: -1.36
95% lower bound for difference: -5.18
T-Test of difference = -3 (vs >): T-Value = 0.74 P-Value = 0.233 DF = 20
Both use Pooled StDev = 5.1956

d)

Two-Sample T-Test and CI: Monthly, Monthly DJIA Return

Two-sample T for Monthly vs Monthly DJIA Return

N Mean StDev SE Mean
Monthly 11 -3.60 6.13 1.8
Monthly DJIA Return 11 -2.25 4.05 1.2

Difference = μ (Monthly) - μ (Monthly DJIA Return)
Estimate for difference: -1.36
95% lower bound for difference: -5.21
T-Test of difference = -3 (vs >): T-Value = 0.74 P-Value = 0.234 DF = 17

e)

Test and CI for Two Variances: Monthly, Monthly DJIA Return

Method

Null hypothesis σ(Monthly) / σ(Monthly DJIA Return) = 1
Alternative hypothesis σ(Monthly) / σ(Monthly DJIA Return) ≠ 1
Significance level α = 0.05

F method was used. This method is accurate for normal data only.

Statistics

95% CI for
Variable N StDev Variance StDevs
Monthly 11 6.131 37.584 (4.284, 10.759)
Monthly DJIA Return 11 4.050 16.404 (2.830, 7.108)

Ratio of standard deviations = 1.514
Ratio of variances = 2.291

95% Confidence Intervals

CI for
CI for StDev Variance
Method Ratio Ratio
F (0.785, 2.918) (0.616, 8.516)

Tests

Test
Method DF1 DF2 Statistic P-Value
F 10 10 2.29 0.207

f)

Test and CI for Two Variances: Monthly, Monthly DJIA Return

Method

Null hypothesis σ(Monthly) / σ(Monthly DJIA Return) = 1
Alternative hypothesis σ(Monthly) / σ(Monthly DJIA Return) > 1
Significance level α = 0.05

Statistics

95% Lower
Bound for
Variable N StDev Variance StDevs
Monthly 11 6.131 37.584 3.973
Monthly DJIA Return 11 4.050 16.404 2.832

Ratio of standard deviations = 1.514
Ratio of variances = 2.291

95% One-Sided Confidence Intervals

Lower Bound Lower Bound
for StDev for Variance
Method Ratio Ratio
Bonett 0.710 0.504
Levene 0.620 0.384

Tests

Test
Method DF1 DF2 Statistic P-Value
Bonett 1 — 1.06 0.151
Levene 1 20 0.77 0.196

thank you...