Question

A random sample of 1001 Americans aged 18-24 years showed that 51% believe their health is extremely important. The interval that is 95% certain to contain the proportion of all 18-24 year olds who believe that their health is extremely important is ________.

A) between 48% and 54%.
B) between 50% and 52%.
C) between 46% and 56%.
D) between 41% and 61%.

Answer 1

A) between 48% and 54%

Explanation:

given that a random sample of 1001 Americans aged 18-24 years showed that 51% believe their health is extremely important.

Here

Sample size n =1001

sample proportion p = 0.51

Standard error of proportion =
`\sqrt{(pq)/(n) } \\=\sqrt{(0.51*0.49)/(1001) } \\=0.0158`

Since sample size is sufficiently large we can take Z critical value for confidence interval

Z critical value for 95% = 1.96

Margin of error = ±1.96*std error

= ±0.0310

confidence interval = 0.51 ±0.0310

= (0.48, 0.54)

i.e. A) between 48% and 54%