Answer:
a)  
    
 
  
If we compare the p value and the significance level given  we see that
 we see that  so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can conclude that the true mean is not different from 410 at 5% of signficance.
 so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can conclude that the true mean is not different from 410 at 5% of signficance.  
b)  
  
Since is a two-sided test the p value would be:  
 
  
Conclusion  
If we compare the p value and the significance level given  we see that
 we see that  so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significantly different from 410 at 5% of signficance.
 so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significantly different from 410 at 5% of signficance.  
Step-by-step explanation:
Part a
Data given and notation  
 represent the sample mean
 represent the sample mean    
 represent the population standard deviation
 represent the population standard deviation 
 sample size
 sample size  
 represent the value that we want to test
 represent the value that we want to test  
 represent the significance level for the hypothesis test.
 represent the significance level for the hypothesis test.  
z would represent the statistic (variable of interest)  
 represent the p value for the test (variable of interest)
 represent the p value for the test (variable of interest)  
State the null and alternative hypotheses.  
We need to conduct a hypothesis in order to check if the true mean differs from 410, the system of hypothesis would be:  
Null hypothesis: 
  
Alternative hypothesis: 
  
Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  
 (1)
 (1)  
z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  
Calculate the statistic  
We can replace in formula (1) the info given like this:  
 
  
P-value  
Since is a two-sided test the p value would be:  
 
  
Conclusion  
If we compare the p value and the significance level given  we see that
 we see that  so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can conclude that the true mean is not different from 410 at 5% of signficance.
 so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can conclude that the true mean is not different from 410 at 5% of signficance.  
Part b
 
  
P-value  
Since is a two-sided test the p value would be:  
 
  
Conclusion  
If we compare the p value and the significance level given  we see that
 we see that  so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significantly different from 410 at 5% of signficance.
 so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significantly different from 410 at 5% of signficance.