At the end of 2018, the variance in the semiannual yields of overseas government bonds was 0.65. A group of bond investors met at that time to discuss future trends in overseas bond yields. Some expected the variability in overseas bond yields to increase, while others took the opposite view. A random sample of 15 overseas bonds taken in March, 2019, generated an average semiannual yield of 3.357 percent with standard deviation 0.721. We want to test if the variance in overseas bond yields has changed since 2018. Use significance level 0.05.

We need to find in the chi square distribution with df = 14 a critical value who accumulates 0.025 of the area and we got the critical value on this case :

If we compare the calculated value is between the two critical values so then we don't have enough evidence to reject the null hypothesis that the true variance is different from 0.65 on this case.

Step-by-step explanation:

Data given

represent the sample size

represent the confidence level  

represent the sample variance obtained

represent the value that we want to test

Null and alternative hypothesis

On this case we want to check if the population variance has changed, so the system of hypothesis would be:

Null Hypothesis:

Alternative hypothesis:

Calculate the statistic  

For this test we can use the following statistic:

And replacing we got:

Calculate the critical value

We need to find in the chi square distribution with df = 14 a critical values who accumulates 0.025 of the area and we got the critical values on this case :

Conclusion

If we compare the calculated value is between the two critical values so then we don't have enough evidence to reject the null hypothesis that the true variance is different from 0.65 on this case.

the first one is x^2+2x+2

the second one is x^2+4

the last one is x^2-4x+8

2(2x + 3) = -4 + 2(2x + 5)

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 6

4x - 6 + 6 = 4x - 6 + 6

4x = 4x

x = x

unlimited solutions.