Question 969825
mean= 0.72 s  sd=0.10 s
z= (1-0.72)/0.10= 0.28/0.10 and it is a one sided z-score of +2.8
P (z>2.8)=0.0026

ii.  That would be a z-score of (0.70-0.72)/.10     or z-score of -0.2  P=0.0793 
     That would be a z-score of (0.8-0.72)/10       or z-score of + 0.8 P=0.2881  ;   Total P=0.3674
b.  90% messages: would be a z-score of >+1.28   or .128 sec or 0.848 (0.85) sec
c.  Sampling distribution
  Probability z < 0.70 sec:  if we get that, we subtract it from 1
z score is- 0.02/[(0.1)/4]    = 0.08/1  z=-0.8  That probability is 0.2881, so probability > 0.70 sec in sampling distribution is 1-0.2881= 0.7119.

Between 0.7 sec and 0.8 sec 
(0.7-0.72)/(0.1/4); that is - 0.08/0.1  or z>-0.8    This is 0.2881 on the left side of the curve
(0.8-0.72)/(0.1/4); that is  +0.08*4/0.1 or z<3.2   This is 0.5000 on the right side
The probability is 0.7881.

90% of the average transmission time will be between z values of +/- 1.645
xb +/- 1.645 (0.1)/4 , where numerical value is 0.041
Rounding, 90% are between 0.68 and 0.76  for the sample.
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2.a z=  (pb-p)/sqrt [(p*(1-p)/55]
z=(0.2-0.3)/sqrt [(.3*.7)/55]
-(0.1)/0.0618  =-1.618  This is about 0.055.  

Not within 3.5% means that sample proportion is >0.335 or <0.265

Same denominator, numerator is 0.035 
z value > or < 0.566  This is 0.2857 x2=0.5714

Same denominator, numerator is now 0.05 and one-way test  shows probability  of 0.21

80% of the time would be a z-value of +/-1.28 ; I multiply that by the denominator 0.0618 and get 0.079 or about 0.08   The interval would be [0.22, 0.38]

Note:  This problem is complicated by the fact that n is small;  >35% of the population is 19.25, so greater than 35% starts with 20, which is really 36.3%.  There are issues with this throughout the problem. If you do it on a calculator, with z-scores, you have no problem, but if you input data that would be required to answer the question, you will get different values.
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This was a long pair of problems.  I wouldn't be surprised if I missed something in this, but trying things a couple of ways, by hand and by calculator, I got things to make sense.