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State the null and alternate hypotheses and find the degree of freedom.

Sue thinks that there is a difference in quality of life between rural and urban living. She collects information from obituaries in newspapers from urban and rural towns in Idaho to see if there is a difference in life expectancy. A sample of 11 people from rural towns give a life expectancy of xr¯=68.4 years with a standard deviation of sr=7.04 years. A sample of 10 people from larger towns give xu¯=74.4 years and su=6.23 years. Does this provide evidence that people living in rural Idaho communities have different life expectancy than those in more urban communities? Use a 5% level of significance.

(a) State the null and alternative hypotheses: (Type ‘‘mu_r″ for the symbol μr , e.g. mu_rnot=mu_u for the means are not equal, mu_r>mu_u for the rural mean is larger, mu_r<mu_u , for the rural mean is smaller. )
H0 = ?

Ha = ?

(b) The degree of freedom is ?

Walter B. | Success-Based Tutor Specializing in Your StudentSuccess-Based Tutor Specializing in Your...
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Null hypothesis: μrural =  μurban
Alternative hypothesis: μrural ≠ μurban

Variance = s12/n1 + s22/n2 = 7.042/11 + 6.232/10 = 8.38689

test statistic = (74.4 - 68.4)/sqrt(8.38689) = 6/2.89601 = 2.072

using a calculator, the p value is calculated as 5.2%

since the p value is greater than 5%, the null hypothesis is not rejected
Andy C. | Math/Physics TutorMath/Physics Tutor
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Null hypothesis: the means are different ---> their sum is NOT zero

Alternative hypothesis: the means are the same ---> their sum is zero

The formulas are found at:
https://onlinecourses.science.psu.edu/stat500/node/50/

Despite the variance are reasonably close, I chose not
to use the pooled test statistic just to play it safe:

X1 = 68.4
s1 = 7.04

X2 = 74.4
S2 = 6.23

degrees of freedom
-----------------
s1^2/n1 + s2^2/n2 =8.034045454545....

C = 0.5608133567941....

The degrees of freedom is 100 / {10*C^2 + (1-c)^2*10}=19.708451741 which is rounded DOWN to 19.

The test statistic is (74.4 - 68.4)/sqrt(8.03045454545454....) = 2.117294095...

Per the T-Distribution table since the sample size is LESS than 30....
For a 1 tailed test you are 95% certain the means are different.
For a 2 tailed test you are 97.5% certain the means are different.