Question 1208010
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Answers:
(a) <font color=red>27.66</font>
(b) <font color=red>155</font>


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Explanation for Part (a)


n = 36 = sample size
xbar = 28.7 = sample mean
s = 3.8 = sample standard deviation


mu = population mean = unknown
sigma = population standard deviation = unknown


Since n = 36 fits the criteria n > 30,  we can use the Z distribution even if we don't know the value of sigma.


At a 90% confidence level, the critical z value is roughly z = 1.645
Use a reference table or stats calculator to determine this.
I'm using this table
<a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
A table like this should be somewhere in the back of your stats textbook. 
Your professor will likely hand out such a table during exams if s/he expects you to use them.
Refer to the bottom row of the table where it lists the confidence levels. Just above "90%" is the value 1.645


E = margin of error for population mean
E = z*s/sqrt(n)
E = 1.645*3.8/sqrt(36)
E = 1.041833333333 approximately


L = lower boundary of confidence interval
L = xbar - E
L = 28.7 - 1.041833333333
L = 27.658166666667
L = <font color=red>27.66</font> which is the answer to part (a).


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Explanation for Part (b)


E = margin of error for population mean
E = z*s/sqrt(n)
E*sqrt(n) = z*s
sqrt(n) = z*s/E
n = (z*s/E)^2


This formula gives us the minimum sample size needed when we specify a desired margin of error.
In this case we have E = 0.6 
The portion that says "with probability equal to 0.95" refers to a 95% confidence interval.
At 95% confidence, the z critical value is roughly z = 1.960 (use a table or stats calculator).


Here are the inputs we'll need
z = 1.960 (approximate)
s = 3.8
E = 0.6


Let's calculate the minimum sample size.
n = (z*s/E)^2
n = (1.960*3.8/0.6)^2
n = 154.090844444444
n = <font color=red>155</font> .... rounding up to nearest integer.
Despite n being much closer to 154, we must round up to 155 to clear the hurdle.


Let's see what happens when we try n = 154.
E = z*s/sqrt(n)
E = 1.960*3.8/sqrt(154)
E = 0.60018 approximately
This is slightly over the 0.6 threshold we want.
The goal is to get E = 0.6 exactly or E < 0.6


Now try n = 155.
E = z*s/sqrt(n)
E = 1.960*3.8/sqrt(155)
E = 0.59824 approximately
We are now under the 0.6 threshold. 



Answer to part (b) is <font color=red>155</font>
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