Question

In a study of cell phone use and brain hemispheric​ dominance, an Internet survey was​ e-mailed to 2455 subjects randomly selected from an online group involved with ears. 931 surveys were returned. Construct a 99​% confidence interval for the proportion of returned surveys.

Answer 1

The 99% confidence interval for the proportion of returned surveys is between 35.378% and 40.4213%, based on 931 responses from 2455 surveyed subjects.

Step-by-step explanation:

To construct a 99% confidence interval for the proportion of returned surveys, we will use the formula for a proportion confidence interval which includes the sample proportion (π), the z-value that corresponds to the desired level of confidence, and the standard error of the proportion.

The sample proportion (π) can be calculated as the number of returned surveys divided by the total number of surveys sent:

π = 931 / 2455 = 0.379

The z-value for a 99% confidence interval is approximately 2.576. The standard error (SE) of π is calculated using the formula:

SE = √(π(1 - π) / n)

SE = √(0.379(1-0.379)/2455)

SE = √(0.379 * 0.621 / 2455)

SE = √(0.2353 / 2455)

SE = √(0.0000958)

SE = 0.009788

Now, we can calculate the margin of error (ME):

ME = z * SE

ME = 2.576 * 0.009788

ME = 0.025213

Finally, the 99% confidence interval (CI) is calculated as:

CI = π ± ME

CI = 0.379 ± 0.025213

CI = [0.353787, 0.404213]

We can be 99% confident that the true proportion of returned surveys falls between 35.378% and 40.4213%.

Answer 2

Answer: 0.355,0.405)

Step-by-step explanation:

Given : Significance level :
`\alpha=1-0.99=0.01`

Number of subjects (n) = 2455

Number of surveys returned = 931

The probability of surveys get return will be :-

`p=(931)/(2455)=0.379226069246\approx0.38`

The confidence interval for proportion is given by :-

`p\pm z_(\alpha/2)*\sqrt{(p(1-p))/(n)}`

`0.38\pm z_(0.005)*\sqrt{(0.38(1-0.38))/(2455)}\\\\=0.38\pm(2.576)0.0098\\\approx0.38\pm0.025=(0.355,0.405)`

Hence, the 99​% confidence interval for the proportion of returned surveys : (0.355,0.405)