Question

A bond analyst is analyzing the interest rates for equivalent municipal bonds issued by two different states.
a) At α = 0.05, is there enough evidence to conclude that there is a difference in the interest rates paid by the two states?
State A:
Sample size 60
Mean interest rate (%) 3.2
Population variance .02
State B:
Sample size 60
Mean interest rate (%) 3.4
Population variance .05

Answer 1

Answer:

$z=\frac{(3.2-3.4)-0}{\sqrt{\frac{0.141^2}{60}+\frac{0.224^2}{60}}}}=-5.85$  

The p value can be calculated with this probability:

$p_v =2*P(z<-5.85)=4.91x10^{-9}$  

The p value for this case is a value very low and near to 0 so then we have enough evidence to reject the null hypothesis and we can conclude that the true means for this case are significantly different

Step-by-step explanation:

Information provided

$\bar X_{1}=3.2$ represent the mean for sample A

$\bar X_{2}=3.4$ represent the mean for sample B  

$\sigma_{1}=\sqrt{0.02}= 0.141$ represent the sample standard deviation for A

$s_{2}=\sqrt{0.05}= 0.224$ represent the sample standard deviation for B

$n_{1}=60$ sample size for the group A

$n_{2}=60$ sample size for the group B  

$\alpha=0.05$ Significance level provided

z would represent the statistic

Hypothesis to test

We want to verify if that there is a difference in the interest rates paid by the two states, the system of hypothesis would be:  

Null hypothesis:$\mu_{1}-\mu_{2}=0$  

Alternative hypothesis:$\mu_{1} - \mu_{2}\neq 0$  

The statistic for this case since we know the population deviations is given by:

$z=\frac{(\bar X_{1}-\bar X_{2})-\Delta}{\sqrt{\frac{\sigma^2_{1}}{n_{1}}+\frac{\sigma^2_{2}}{n_{2}}}}$ (1)  

Replacing the info given we got:

$z=\frac{(3.2-3.4)-0}{\sqrt{\frac{0.141^2}{60}+\frac{0.224^2}{60}}}}=-5.85$  

The p value can be calculated with this probability:

$p_v =2*P(z<-5.85)=4.91x10^{-9}$  

The p value for this case is a value very low and near to 0 so then we have enough evidence to reject the null hypothesis and we can conclude that the true means for this case are significantly different