Question 1207859
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Please ask only one question per post.
I'll do problem 1 to get you started.


n = 15 = sample size
xbar = 50.50 = sample mean
s^2 = 400 which leads to s = 20 = sample standard deviation
sigma = population standard deviation = unknown


We don't know the value of sigma and n > 30 is not the case, so we must use the T distribution.
df = degrees of freedom
df = n-1
df = 15-1
df = 14


We'll need to find the t critical value. 
To do so, you could use a stats calculator like a TI83. 
However, I'll use a T table such as this
<a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
Such a table can be found at the back of your stats textbook.
Highlight the df = 14 row and the column that mentions "confidence level = 95%" (mentioned at the bottom of the table)
The intersection of this row and column yields the approximate t critical value t = 2.145


What does this tell us?
It tells us that P(-2.145 < t < 2.145) = 0.95 approximately when df = 14.
The 0.95 is the area of the main body while 1-0.95 = 0.05 is the combined area of the two tails.
The 0.95 refers to the confidence level 95%.


E = margin of error
E = tCritical*s/sqrt(n)
E = 2.145*20/sqrt(15)
E = 11.08 approximately


The confidence interval is centered at xbar. 
The radius of the interval is the margin of error. 
We'll add and subtract (i.e. plus minus) the value of E to xbar so we can determine the boundaries.
L = lower bound = xbar-E
U = upper bound = xbar+E
confidence interval = xbar ± E = <font color=red>50.50 ± 11.08</font>


The final answer is <font color=red>option D</font>
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