Respuesta :
Answer: (i) Interval for proportion of underpaid nurses: 0.15 ± 0.0415.
(ii) Interval for difference in proportion: 0.03 ± 0.0641
Step-by-step explanation: Confidence Interval for a representation of the proportion is calculated by:
[tex]p_{hat}[/tex] ± z.[tex]\sqrt{\frac{p_{hat}(1-p_{hat})}{n} }[/tex]
(i) [tex]p_{hat}[/tex] is the proportion, in this case, of underpaid nurses: [tex]p_{hat}[/tex] = 0.15
Confidence Interval is 90%, so z = 1.645
The interval is:
0.15 ± 1.645.[tex]\sqrt{\frac{0.15(0.85)}{200} }[/tex]
0.15 ± 1.645*0.0252
0.15 ± 0.0415
Which means that, in 90% of times, the proportion of underpaid nurses wil be between 0.1085 and 0.1915.
(ii) For the difference in proportions:
[tex]p_{hat_1}-p_{hat_2}[/tex] ± z.[tex]\sqrt{\frac{p_{hat_1}(1-p_{hat_1})}{n_{1}}+\frac{p_{hat_2}(1-p_{hat_2})}{n_{2}} }[/tex]
The proportion for the second hospital is [tex]p_{hat_2}[/tex] = 0.12 and, since it is the same confidence interval, z = 1.645.
Calculating:
(0.15-0.12) ± 1.645.[tex]\sqrt{\frac{0.15(0.85)}{200} + \frac{0.12(0.88)}{120} }[/tex]
0.03 ± 1.645*0.0389
0.03 ± 0.0641
The 90% confidence interval for the difference in proportions of nurses being underpaid is between 0.0341 and 0.0941
