PLEASE HELP. According to a recent study by the Annenberg foundation, only 36%...

Question

According to a recent study by the Annenberg foundation, only 36% of adults in the United States could name all three branches of government. This was based on a survey given to a random sample of 1416 US adults.

[a] What is the standard deviation of the sample?

a) 3.6%

b) 3%

c) 3.2%

d) 40%

[b] What is the critical value for a 90% confidence interval?

a) 1.645

b) 2.56

c) 1.5

d) 3.54

[c] Given a critical value of 1.645 for a 90% confidence interval and a standard deviation of .032, what is the confidence interval for the point estimate of 36% from the sample of 1416?

a) 31.76% to 42.25%

b) 36.25% to 36.75%

c) 40.25% to 44.25%

d) 36.45% to 44.45%

Answer 1

For a sample proportion, the standard error is:

SE = √[p(1 − p) / n]

Here:

p = 0.36

n = 1416

So:

SE = √[(0.36)(0.64) / 1416]

SE ≈ 0.01276

or about 1.28%.

Therefore, for part (a), none of the listed answers is mathematically correct.

For part (b), a 90% confidence interval uses the critical z-value:

z = 1.645*

So the answer is:

(b) a) 1.645

For part (c), the question specifically tells us to use a standard deviation of 0.032. Using that value:

Margin of error = 1.645(0.032)

= 0.05264

Now add and subtract this from 0.36:

Lower limit = 0.36 − 0.05264 = 0.30736

Upper limit = 0.36 + 0.05264 = 0.41264

Therefore:

30.74% to 41.26%

Again, none of the choices in part (c) matches the stated numbers.

If we instead use the correct standard error based on the sample size of 1416, the 90% confidence interval would be approximately:

0.36 ± 1.645(0.01276)

= 0.36 ± 0.02099

= 33.90% to 38.10%

So there are likely typographical errors in parts (a) and (c) of the question.