Question

3. American Journal of Public Health wanted to investigate the proportion of all elderly who use mental health services in nursing homes. In a study, 4646 nursing home residents were randomly selected and found that 109 of these residents used a mental health specialist. Describe the parameter we are estimating. Find and interpret a 90% confidence interval for the proportion of all elderly who use mental health services in nursing homes. Show all steps to receive full credit.

Answer 1

Solution :

Given that,

n = 4646

x = 109

Point estimate = sample proportion = = x / n = 109/4646=0.023

1 -   = 1- 0.023 =0.977

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z0.05 = 1.645 ( Using z table )

Margin of error = E Z/2 *(( * (1 - )) / n)

= 1.645 *((0.023*0.977) / 4646)

E = 0.0036

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.023-0.0036 < p < 0.023+0.0036

0.0194< p < 0.0266

The 90% confidence interval for the population proportion p is : 0.0194,0.0266