MATH SOLVE

4 months ago

Q:
# Andrew thinks that people living in a rural environment have a healthier lifestyle than other people. He believes the average lifespan in the USA is 77 years. A random sample of 11 obituaries from newspapers from rural towns in Idaho gives x¯=79.14 and s=2.48. Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years? Use a 10% level of significance.

Accepted Solution

A:

Answer:Yes, it does.Step-by-step explanation:Since the data is approximately Normally distributed and the sample is less than 30, we will be using the Student's t Distribution.
[tex] \bf H_0[/tex]: The life expectancy of people living in rural Idaho is 77 years
[tex] \bf H_a[/tex]: The life expectancy of people living in rural Idaho is greater than 77 years.
So, this is a right-tailed test.
Our t-statistic is given by
[tex] \bf t=\frac{\bar x-\mu}{s/\sqrt{n}}[/tex]
where
[tex] \bf \bar x[/tex] = 79.14 is the mean of the sample
[tex] \bf \mu[/tex] = 77 is the mean of the null hypothesis
s = 2.48 is the sample standard deviation
n = 11 is the sample size
Computing our t-statistic we get
[tex] \bf t=\frac{79.14-77}{2.48/\sqrt{11}}=2.862[/tex]
Now, we obtain the critical upper value [tex] \bf t^*[/tex] for a right-tailed test hypothesis corresponding to a 10% level of significance associated with the Student's t Distribution with 10 degrees of freedom (sample size-1). This is a value [tex] \bf t^*[/tex] such that the area under the t distribution to the left of [tex] \bf t^*[/tex] equals 10% = 0.01
We can do it either by looking up a table or a spreadsheet.
In Excel use
TINV(0.2,10)
In OpenOffice Calc use
TINV(0.2;10)
and we would get [tex] \bf t^*[/tex] = 1.3722
Since our t-statistic is greater than [tex] \bf t^*[/tex] we can reject the null hypothesis and say this sample provide evidence that people living in rural Idaho communities live longer than 77 years.