Question

A Pew Research study of 4726 randomly selected U.S. adults regarding scientific human enhancements, found that approximately 69% of the sample stated that they were worried about brain chip implants being used for improving cognitive abilities.

Required:
a. Show that the necessary conditions (Randomization Condition, 10% Condition, Sample Size Condition) are satisfied to construct a confidence interval. Briefly explain how each condition is satisfied.
b. Find the 90% confidence interval for the proportion of all U.S. adults that are worried about brain chip implants used for improving cognitive abilities.
(To show your work: Write down what values you are entering into the confidence interval calculator.)
c. Briefly describe the meaning of your interval from part (b).

Answer 1

a)Randomization condition: Satisfied, as the subjects were randomly selected.

10% condition: Satisfied, as the sample size is less than 10% of the population (U.S. adults).

Sample size condition: Satisfied, as the product between the smaller proportion and the sample size is bigger than 10.

b) The 90% confidence interval for the population proportion is (0.68, 0.70).

Explanation:

a) Evaluating the necessary conditions:

Randomization condition: Satisfied, as the subjects were randomly selected.

10% condition: Satisfied, as the sample size is less than 10% of the population (U.S. adults).

Sample size condition: Satisfied, as the product between the smaller proportion and the sample size is bigger than 10.

`n(1-p)=4,726\cdot (1-0.69)=4,726\cdot 0.31=1,465>10`

b) We have to calculate a 90% confidence interval for the proportion.

The sample proportion is p=0.69.

The standard error of the proportion is:

`\sigma_p=\sqrt{(p(1-p))/(n)}=\sqrt{(0.69*0.31)/(4726)}\\\\\\ \sigma_p=√(0.000045)=0.007`

The critical z-value for a 90% confidence interval is z=1.645.

The margin of error (MOE) can be calculated as:

`MOE=z\cdot \sigma_p=1.645 \cdot 0.007=0.01`

Then, the lower and upper bounds of the confidence interval are:

`LL=p-z \cdot \sigma_p = 0.69-0.01=0.68\\\\UL=p+z \cdot \sigma_p = 0.69+0.011=0.70`

The 90% confidence interval for the population proportion is (0.68, 0.70).