Answer:
a) 
And for this case if we use this formula we got:
b) Since we have n =16 values for the sample the median can be calculated as the average between position 8th anf 9th and we got:
c) 
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 90% of data from the top 10% is 1.8024.
d) 
The variance for this estimator is given by:
We can assume the obervations independent so then we have:
And replacing we got:
And the standard error would be given by:
Step-by-step explanation:
Data given:
0.86 0.88 0.88 1.07 1.09 1.17 1.29 1.31 1.46 1.49 1.59 1.62 1.65 1.71 1.76 1.83
Part a
We can calculate the mean with the following formula:
And for this case if we use this formula we got:
Part b
For this case in order to calculate the median we need to put the data on increasing way like this:
0.86 0.88 0.88 1.07 1.09 1.17 1.29 1.31 1.46 1.49 1.59 1.62 1.65 1.71 1.76 1.83
Since we have n =16 values for the sample the median can be calculated as the average between position 8th anf 9th and we got:
Part c
For this case we can assume that the mean is 
And we can calculate the population deviation with the following formula:
And if we replace we got: 
And assuming normal distribution we have this:
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 90% of data from the top 10% is 1.8024.
Part d
The median is defined as :
The variance for this estimator is given by:
We can assume the obervations independent so then we have:
And replacing we got:
And the standard error would be given by:
