Question

According to the historical data, the life expectancy in Argentina is equal to the life expectancy in Bolivia. A new study has been made to see whether this has changed. Records of 265 individuals from Argentina who died recently are selected at random. The 265 individuals lived an average of 74.8 years with a standard deviation of 4.1 years. Records of 300 individuals from Bolivia who died recently are selected at random and independently. The 300 individuals lived an average of 75.4 years with a standard deviation of 4.3 years. Assume that the population standard deviation of the life expectancy can be estimated by the sample standard deviations, since the samples that are used to compute them are quite large. At the 0.05 level of significance, is there enough evidence to support the claim that the life expectancy, μ1, in Argentina is not equal to the life expectancy, μ2, in Bolivia anymore? Perform a two-tailed test. Then fill in the table below.
Carry your intermediate computations to at least three decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

The null hypothesis: H sub 0:

The alternative hypothesis: H sub 1:

The type of test statistic:

The value of the test statistic:

The two critical values at the 0.05 level of significance:

Can we support the claim that the life expectancy in Argentina is not equal to the life expectancy in Bolivia? Yes or No

Answer 1

Answer:

Step-by-step explanation:

Hello!

The historical data suggests that the life expectancy in Argentina is equal to the life expectancy in Bolivia.

With the objective of testing if that it hasn't changed, the records of recently deceased people from Argentina and Bolivia were selected at random:

Group 1: Argentina

X₁: Years of life of a recently deceased Argentinian.

n₁= 265

X[bar]₁= 74.8 years

S₁= 4.1 years

Group 2: Bolivia

X₂: Years of life of a recently deceased Bolivian.

n₂= 300

X[bar]₂= 75.4 years

S₂= 4.3 years

The parameters of interest are the population means of the years of life of people in both countries:

H₀: μ₁ = μ₂

H₁: μ₁ ≠ μ₂

α: 0.05

The statistic to use is an approximate standard deviation, and since both samples are quite large, it is valid to use the sample standard deviations in place of the population standard deviations:

$Z= \frac{(X[bar]_1-X[bar]_2)-(Mu_1-Mu_2)}{\sqrt{\frac{S^2_1}{n_1} +\frac{S_2^2}{n_2} } }$≈N(0;1)

$Z_{H_0}= \frac{(74.8-75.4)-(0)}{\sqrt{\frac{16.81}{265} +\frac{18.49}{300} } }= -2.54$

The critical values for this test are:

$Z_{\alpha /2}= Z_{0.025}= -1.96$

$Z_{1-\alpha /2}= Z_{0.0975}= 1.96$

Using the critical value approach, the decision rule is:

If $Z_{H_0}$ ≤ -1.96 or if $Z_{H_0}$ ≥ 1.96, reject the null hypothesis.

If -1.96 < $Z_{H_0}$ < 1.96, do not reject the null hypothesis.

$Z_{H_0}$ ≤ -1.96 so the decision is to reject the null hypothesis.

So with a 5% significance level, there is enough evidence to reject the null hypothesis, you can conclude that the life expectancy in Argentina is different from the life expectancy in Bolivia.

I hope this helps!