Question

Find confidence interval and perform a hypothesis test for the mean of one population.

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 #1. 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 #1

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?

b) What are the assumptions to run the test?

c) What is the null hypothesis?

d) What is the alternative hypothesis?

e) Determine if this test is left-tailed, right-tailed, or two-tailed.

f) What is the significance level?

g) What is the test statistics?

h) What is the p-value?

i) Do we reject the null hypothesis? Why?

j) What is the conclusion?

k) What is 95% confidence interval for the population mean?

l) Interpret the confidence interval.

Answer 1

= 43.8726

s = 9.0712

n = 26

a) Since the population standard deviation is unknown, so we will use t-test.

b) We will asuume that the population from which the samples are drawn is normally distributed.

c) H₀: = 60.29

d) H₁: < 60.29

e) This is a left-tailed test

f) Significance level is 0.05

g) The test statistic t = ()/(s/)

                                 = (43.8726 - 60.29)/(9.0712/)

                                 = -9.228

h) P-value = P(T < -0.228)

                 = 0.000

i) Since the P-value is less than the significance level (0 < 0.05), so we should reject the null hypothesis.

j) So the data provides sufficient evidence to conclude that the mean economic dynamism of middle-income countries is less than the mean for high income countries.

k) At 95% confidence interval the critical value is t^* = 2.06

The 95% confidence interval for population mean is

+/- t^* * s/

= 43.8726 +/- 2.06 * 9.0712/

= 43.8726 +/- 3.665

= 40.2076, 47.5376

l) WE are 95% confident that the true population mean lies in the above interval.