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 provided below ("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. 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

a) What is the appropriate test for this case? (10 points)

b) What is the null hypothesis? (5 points)

c) What is the alternative hypothesis? (5 points)

d) Determine if this test is left-tailed, right-tailed, or two-tailed. (5 points)

e) What is the significance level? (5 points)

f) What is the test statistic? (10 points) g) What is the p-value? (10 points)

h) Do we reject the null hypothesis? (5 points)

i) What is the conclusion? (15 points)

j) What is 95% confidence interval for the population mean? (10 points)

k) Interpret the confidence interval. (15 points)

Answer 1

Here, we have given that

Claim: To check whether the mean economic dynamism of middle-income countries is less than the mean for high-income countries i.e. 60.29.

(a) One sample t test is appropriate test for above claim

The Hypothesis is as follows

(b)

The null hypothesis is as follows

v/s

(c) The alternative hypothesis is as follows

(d)

This is the left tailed test

(e)

= level of significance=0.05

(f)

We have given that,

Xi: Economic dynamism of middle income countries

Xi
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

n= Number of observation = 26

= sample mean =43.87

S= Sample standard deviation =9.07

Now, we can find the test statistic

t-statistics=

=

= -9.230

we get,

the Test statistic is -9.230

(g)

Now we find the P-value

= level of significance=0.05

This is left i.e. one  tailed test

Degrees of freedom =n-1=26-1=25

Now, we can find the P-value

P-value = 0.0001 (using EXECl =TDIST( t-stat= 9.230, D.F=25, tail=1)

we get the P-value is 0.0001

(h)

Decision:

P-value < 0.05 ()

That is we reject Ho (Null Hypothesis)

(i)

Conclusion

There is the sufficient evidence to conclude that the mean economic dynamism of middle-income countries is less than the mean for high-income countries i.e. 60.29.

(j)

Now, we want to find the 95% confidence interval for population mean

Formula is as follows,

Where

E=Margin of error =

Now,

Degrees of freedom = n-1 = 26-1=25

c=confidence level =0.95

=level of significance=1-c=1-0.95=0.05

and we know that confidence interval is always two tailed

t-critical = 2.060 ( using t table )

Now,

=

=3.664

We get the 95% confidence interval for the population mean

(k)

Interpretation:

This is the 95% CI which shows that we have 95% confidence that this population mean will fall within this interval.