Question

7. A random sample of 20 stock return is believed to be normally distributed with mean u and variance ơ2. The returns. X. are recorded as follows: 0.03 0.090 0.022 0.100 0.0120.000.0160.1310.0380.038 0.107 0.165 0.102 0.0060.047 0.010 0.0710.094 0.029 0.057 By setting α-0.10, test the hypothesis Ho: σ2 0.01 against the alternative, H1:02 < 0.01 Determine the 95% confidence intervals for by assuming that, a. b. Ơ2 0.0 1, and σ2 is not known. I. 11,

Answer 1

7)

	stock return
	0.031
	0.09
	0.022
	0.1
	0.012
	0.001
	0.016
	0.131
	0.038
	0.038
	0.107
	0.165
	0.102
	0.006
	0.047
	0.01
	0.071
	0.094
	0.029
	0.057
sample mean	0.05835
sample std dev	0.046617452
sample variance	0.002173187

a)

The Chi-Square test for one population variance is used to test whether the sample variance is greater or less than 0.01. The test is performed in following steps,

Step 1:

The null and alternative hypotheses are,

Но : σ2 > 0.01

H1 : σ2 < 0.01

Step 2: decision rule,

The critical value for the chi square statistic is obtained from chi square distribution table for significance level = 0.05 and degree of freedom = n -1 = 20 - 1 = 19.

critical10.117

Since this is left tailed test, reject the null hypothesis if,

X-< crític,-10. 117

Step 3: The Chi-Squared statistic is

2 (n -1)s2 (20 1)0.00217 = 4.129 0.01

Step 4:

Since,

2 = 4. 129 critical < eri ticalー10.117

The null hypothesis is rejected.

b)

i)

For known population variance z statistic critical value and population variance is used to calculate the confidence interval as shown below,

$\text{95\% CI}=\overline{X}\pm z_c \times \frac{\sigma}{\sqrt{n}}$

For 95% CI, z_c = 1.96

0.1 95% CI = 0.05835± 1.96 × V20

95% CI 0.05835 0.044

% CI= (0.015, 0.102

ii)

For unknown population variance t statistic critical value and sample variance is used to calculate the confidence interval as shown below,

$\text{95\% CI}=\overline{X}\pm t \times \frac{s}{\sqrt{n}}$

For 95% CI and degree of freedom = n - 1 = 20 - 1 = 19, t = 2.093

0.0466 95% CI = 0.05835 ± 2.093 ×

$\text{95\% CI}=0.05835\pm 0.022$

95'% CI = (0.037, 0.08)