Question

1 point) Ann 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 10 people from rural towns give a life expectancy of xr¯=78.5xr¯=78.5 years with a standard deviation of sr=6.75sr=6.75 years. A sample of 3 people from larger towns give xu¯=74.4xu¯=74.4 years and su=7.57su=7.57 years. Does this provide evidence that people living in rural Idaho communities have different life expectancy than those in more urban communities? Use a 1% 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, Ann 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 :

a) The null and alternative hypotheses are as follows :

b) The degrees of freedom is given by,

We have,

The degrees of freedom is 11.

c) To test the hypothesis we shall use two samples t-test assuming equal population variances. The test statistic is given as follows :

Where,

We have,  

Also we have,

The value of the test statistic is 0.9018.

d) Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic. The two-tailed p-value is given as follows :

p-value = 2.P(T > |t|)

We have, |t| =0.9018

p-value = 2.P(T > 0.9018)

p-value = 0.3865

The p-value is 0.3865.

Significance level = 1% = 0.01

(0.3865 > 0.01)

Since, p-value is greater than the significance level of 0.01, therefore we shall be fail to reject the null hypothesis (H​​​​​​​​​​​0) at 1% significance level.

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

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