Question 1202151
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Answer:  <font color=red size=4>(97.26, 98.80)</font>
That is the condensed form of 97.26 < mu < 98.80



Work Shown:


mu = μ = Greek letter representing population mean


Specifically mu is the population mean high temperature.
The goal is to estimate mu using a confidence interval.


Given data set = {96.5,98.7,99.4,97,97.9,98.9,97.8}
n = 7 = sample size
xbar = sample mean
xbar = (add up the values)/(number of values)
xbar = (96.5+98.7+99.4+97+97.9+98.9+97.8)/(7)
xbar = 98.029 approximately



Use a calculator or spreadsheet to determine the sample standard deviation is approximately s = 1.045


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*1.045/sqrt(7)
E = 0.767
That result is approximate.


Then,
L = lower boundary
L = xbar - E
L = 98.029 - 0.767
L = 97.262
L = 97.26
and
U = upper boundary
U = xbar + E
U = 98.029 + 0.767
U = 98.796
U = 98.80



The 90% confidence interval in the format L < mu < U is roughly 97.26 < mu < 98.80


That is then condensed to the format (L, U) so we get <font color=red size=4>(97.26, 98.80)</font> as the final answer.


We are 90% confident that the population mean high temperature is somewhere between 97.26 degrees Fahrenheit and 98.80 degrees Fahrenheit.
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