Question

10.31 Does age make a difference in the amount of savings a worker feels is needed to be secure in retirement? A study by the Association of Superannuation Funds of Australia (ASFA) in 2020 found that 0.26 of workers aged 25-33 years feel that $744 000 for couples is enough to be secure at retirement. However, 0.35 of workers aged 34-52 years feel that this amount is enough. Suppose 124 workers aged 25-33 years and 146 workers aged 34-52 years were involved in this study. Use these data to construct a 90% confidence interval to estimate the difference between population proportions on this question.

Answer 1

10.31

The confidence interval is given by

P1 (1- P1), P2(1- p2) +*V P1 (1 – P1) p2 (1 – p2) pi – p2 – , рі — р2 + zc) N2 N1 N2 N1

where,

$\hat p_1=0.26$$\hat p_2=0.35$$N_1=124$$N_2=146$

Level of significance, $\alpha=1-0.9=0.1$

So, critical value _{$z_c=z_{1-\alpha/2}=z_{0.95}=1.645$}

So, 90% confidence interval is given by

$\left ( 0.26-0.35 -1.645\sqrt{\frac{0.26\left ( 1-0.26\right )}{124}+\frac{0.35\left ( 1-0.35 \right )}{146}} , 0.26-0.35 +1.645\sqrt{\frac{0.26\left ( 1-0.26\right )}{124}+\frac{0.35\left ( 1-0.35 \right )}{146}} \right )$

i.e. (-0.182,0.002).

So, based on the given data we can obtain a 90% confidence interval as (-0.182,0.002).