Question

Mike thinks that there is a difference in quality of life between rural and urban living. He 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¯=73.7 years with a standard deviation of sr=5.54 years. A sample of 16 people from larger towns give xu¯=78.4 years and su=8.49 years. Does this provide evidence that people living in rural Idaho communities have different life expectancy than those in more urban communities? Use a 10% level of significance.

(a) State the null and alternative hypotheses: (Type ‘‘mu_r″‘‘mu_r″ for the symbol μrμr , e.g.  mu_rnot=mu_umu_rnot=mu_u for the means are not equal, mu_r>mu_umu_r>mu_u for the rural mean is larger,  mu_r<mu_umu_r<mu_u , for the rural mean is smaller. )
H0H0 =  
HaHa =

(b) The degree of freedom is

(c) The test statistic is

(d) Based on this data, Mike concludes:

A. There is not sufficient evidence to show that life expectancies are different for rural and urban communities.  
B. The results are significant. The data seems to indicate that people living in rural communities have a different life expectancy than those in urban communities.

Answer 1

The statistical software output for this problem is:

Two sample T summary hypothesis test:
μ₁ : Mean of Population 1
μ₂ : Mean of Population 2
μ₁ - μ₂ : Difference between two means
H₀ : μ₁ - μ₂ = 0
H_A : μ₁ - μ₂ ≠ 0
(with pooled variances)

Hypothesis test results:

Difference	Sample Diff.	Std. Err.	DF	T-Stat	P-value
μ1 - μ2	-4.7	2.9185602	25	-1.6103831	0.1199

Hence,

a) Ho = mu_r = mu_u

Ha = mu_r not= mu_u

b) Degrees of freedom = 25

c) Test statistic = -1.610

d) Option A is correct.