Question

Question 4 [10 Marks]

The table below shows the percentage of population below poverty level in ten localities, as determined in the years 1990 and 1998.

Locality	1	2	3	4	5	6	7	8	9	10
1998	11.6	9.8	10.1	8.7	20.4	11.2	13.1	10.4	16.6	13.7
1990	7.5	13.4	13.7	10.7	20.9	11.5	14.4	13.1	13.7	16.2

1. Is this paired or unpaired data?
2. Calculate the summary statistics relevant to this problem.
3. Compute a 95% confidence interval for the mean change in poverty levels. Show all your working.
4. Based on that confidence interval, is there sufficient evidence to conclude that the poverty level changed overall been 1990 and 1998?
5. The investigators want to estimate the difference in poverty level to an accuracy of ±1% between 1990 and 1998 with 95% confidence. How many measurements would be needed to achieve this?

Answer 1

(a)

This is paired data , because for the same locality poverty level is measured twice: in 1998 and 1990.

(b_

Summary statistics is got as follows:

Values of d = Poverty level of 1998 - Poverty level of 1990 are got as follows:

d = 4.1 - 3.6, - 3.6, - 2.0, - 0.5, -0.3, -1.3, - 2.7, 2.9, - 2.5

From d values, the following statistics are calculated:

n = 10

= - 9.5/10 = - 0.95

s_d = 2.6180

SE = s_d/

= 2.6180/

= 0.8279

(c)

=0.05

ndf = n - 1 = 10 - 1 = 9

From Table, critical values of t = 2.2622

Confidence interval:

- 0.95 (2.2622 X 0.8279)

= - 0.95 1.8728

= ( - 2.8228 , 0.9228)

Confidence Interval:

- 2.8228 < < 0.9228

(d)

Based on that confidence interval, there is no sufficient evidence to conclude that the poverty level changed overall between 1990 and 1998 because 0 is included in the confidence interval.

(e)

Sample Size (n) is given by:

Given:

= 0.05

From Table, critical values of Z = 1.96

= 2.6180

e = 0.01

Substituting, we get:

= 263301

So,

Answer is:

263301