Question

The economic dynamism, which is the index of productive growth in dollars for countries that are designated by the World Bank as middle-income are in table #7.3.8 ("SOCR data 2008," 2013). Countries that are considered high-income have a mean economic dynamism of 60.29. Do the data show that the mean economic dynamism of middle-income countries is less than the mean for high-income countries? Test at the 5% level. Table #7.3.8: Economic Dynamism of Middle Income Countries

25.8057 37.4511 51.915 43.6952 47.8506 43.7178 58.0767 41.1648 38.0793 37.7251 39.6553 42.0265 48.6159 43.8555 49.1361 61.9281 41.9543 44.9346 46.0521 48.3652 43.6252 50.9866 59.1724 39.6282 33.6074 21.6643

Answer 1

No. At a significance level of 0.05, there is not enough evidence to support the claim that the mean economic dynamism of middle-income countries is less than the mean for high-income countries (60.29).

Test staitistic t= -1.02

P-value=0.159

Explanation:

We have a sample of size n=26, with mean 43.8727 and standard deviation s=82.2857.

`M=(1)/(n)\sum_(i=1)^n\,x_i\\\\\\M=(1)/(26)(25.8057+37.4511+51.915+43.6952+47.8506+. . .+21.6643)\\\\\\M=(1140.689)/(26)\\\\\\M=43.8727\\\\\\s=(1)/(n-1)\sum_(i=1)^n\,(x_i-M)^2\\\\\\s=(1)/(25)((25.8057-43.8727)^2+. . . +(21.6643-43.8727)^2)\\\\\\s=(2057.1431)/(25)\\\\\\s=82.2857\\\\\\`

This is a hypothesis test for the population mean.

The claim is that the mean economic dynamism of middle-income countries is less than the mean for high-income countries (60.29).

Then, the null and alternative hypothesis are:

`H_0: \mu=60.29\\\\H_a:\mu< 60.29`

The significance level is 0.05.

The sample has a size n=26.

The sample mean is M=43.8727.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=82.2857.

The estimated standard error of the mean is computed using the formula:

`s_M=(s)/(√(n))=(82.2857)/(√(26))=16.138`

Then, we can calculate the t-statistic as:

`t=(M-\mu)/(s/√(n))=(43.8727-60.29)/(16.138)=(-16.42)/(16.138)=-1.02`

The degrees of freedom for this sample size are:

`df=n-1=26-1=25`

This test is a left-tailed test, with 25 degrees of freedom and t=-1.02, so the P-value for this test is calculated as (using a t-table):

`\text{P-value}=P(t<-1.02)=0.159`

As the P-value (0.159) is bigger than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the mean economic dynamism of middle-income countries is less than the mean for high-income countries (60.29).