Question

(1 point) 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 20 people from rural towns give a life expectancy of xr¯=80.8xr¯=80.8 years with a standard deviation of sr=9.54sr=9.54 years. A sample of 4 people from larger towns give xu¯=76xu¯=76 years and su=9.44su=9.44 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

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

a)

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = 5.17967

b)

D.F = n₁ + n₂ - 2
DF = 22
c)

t = [ (x₁ - x₂) - d ] / SE

t = 0.9267

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 22 degrees of freedom is more extreme than - 0.9267; that is, less than -0.9267 or greater than 0.9267.

P-value = P(t < - 0.9267) + P(t > 0.9267)

Use the t-value calculator for finding p-values.

P-value = 0.1821 + 0.1821

P-value = 0.3642

Interpret results. Since the P-value (0.3642) is greater than the significance level (0.10), we cannot reject the null hypothesis.

d)

A. There is not sufficient evidence to show that life expectancies are different for rural and urban communities.