Question

1.) It is reported that 61% of Americans feel that the president is doing an acceptable job. A random sample of 100 people is taken. What is the probability that the proportion of Americans in the sample feel that the president is doing an acceptable job is less than 0.5?

Correct answer: 0.0121

Explain how to get to the answer above.

2.) A telephone poll of 1000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” 20% answered “crime”. Find a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.

Correct answer: (0.175 , 0.225)

Explain how to get to the answer above.

Answer 1

Solution

Given that,

p = 0.61

1 - p = 1 - 0.61=0.39

n = 100

= p =0.61

=  [p ( 1 - p ) / n] =   [(0.61*0.39) / 100 ] = 0.0488

P( <0.5 ) =

= P[( - ) / < (0.5 -0.61) / 0.0488]

= P(z < -2.25)

Using z table,

= 0.0121

(B)

Solution :

Given that,

n = 1000

Point estimate = sample proportion = = 0.20

1 - = 1 - 0.20=0.80

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.96}

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

= 1.96 (((0.20*0.80) /1000 )

= 0.025

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.20 - 0.025< p < 0.20+0.025

0.175< p <0.225

The 95% confidence interval for the population proportion p is :0.175 , 0.225