Question 1081012
The standard error is sqrt(.6*.4/50)=0.0693
It is a two-sided test, and the z value is 1.645
The distribution of the mean is (0.60) with SE 0.0693 and 95% CI interval is +/- 0.135
Note, for 90% CI the interval is +/- 0.1140
I am not clear on some of the questions, such as verify if the means are different when variance and sample sizes are known. 
At the 90% level, the confidence interval is (0.486, 0.714) and it is even wider at the 95% level.  The CI contains the parameter 0.55 or 55%, so fail to reject Ho.  If Ho should have been rejected, but we accepted a false null hypothesis, that is a Type II error.  Here, we fail to reject Ho at both the 5 and 10% levels, and the p-value for this is 0.48.

Note: the z-test starts with the assumption that there is no difference.  The z-test for a two way test with 0.10 significance is >|1.645| The critical values are set which, if exceeded, say that the value that was found was so not likely to be due to chance that the null hypothesis should be rejected. The p-value is the likelihood that if the null hypothesis were true, the likelihood of finding a result or greater (in either direction away from Ho) is in this instance 0.48. Low p-values say that the change is highly unlikely to be due to random factors.

Also note that we are looking for a difference of a proportion from a hypothesized proportion.  It happens to be the case that the mean and the variance in a proportion test are not independent, which means if you know the mean, you also know the variance, unlike a Gaussian or normal distribution.