Question 1202152
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Answer:  <font color=red size=4>(97.169, 98.374)</font>
That is the condensed form of 97.169 < mu < 98.374



Work Shown:



mu = μ = Greek letter representing population mean


In this problem's context, mu is the population mean body temperature in degrees Fahrenheit.
The goal is to estimate mu using a confidence interval.


Given data set = {97.1,98.1,98,97.7,97.4,99.3,96.8}
n = 7 = sample size
xbar = sample mean
xbar = (add up the values)/(number of values)
xbar = (97.1+98.1+98+97.7+97.4+99.3+96.8)/(7)
xbar = 97.77143 approximately



Use a calculator or spreadsheet to determine the sample standard deviation is approximately s = 0.81999 


df = degrees of freedom = n-1 = 7-1 = 6


Because the population standard deviation (sigma) is not known, and because n > 30 isn't the case, we must use the T distribution.


Refer to this T table
<a href = "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
Locate the row labeled df = 6
Locate the column labeled "90% confidence". The confidence labels are at the bottom. 
The approximate value t = 1.943 is at this row and column intersection. It is the t critical value.


Specialized stats calculators such as this one
<a href = "https://www.omnicalculator.com/statistics/critical-value">https://www.omnicalculator.com/statistics/critical-value</a>
can find the t critical value. 
Make sure to do a two-tailed test. 
Also set the significance level to 0.10 (recall that alpha = 1-C where C is the confidence level)


Compute the margin of error
E = t*s/sqrt(n)
E = 1.943*0.81999/sqrt(7)
E = 0.60219
That result is approximate.


Then,
L = lower boundary
L = xbar - E
L = 97.77143 - 0.60219
L = 97.16924
L = 97.169
and
U = upper boundary
U = xbar + E
U = 97.77143 + 0.60219
U = 98.37362
U = 98.374



The 90% confidence interval in the format L < mu < U is roughly 97.169 < mu < 98.374


That is then condensed to the format (L, U) so we get <font color=red size=4>(97.169, 98.374)</font> as the final answer.


We are 90% confident that the population mean temperature is somewhere between 97.169 degrees Fahrenheit and 98.374 degrees Fahrenheit.
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