Question

A study was done on the effectiveness of lie detector tests to catch someone that lies. In a random sample of 48 total lies, the machine identified only 31 of them. What was the sample percentage? Now construct a 99% confidence interval estimate of the population percentage of lies caught by the lie detector machine.

Answer 1

Answer 2

To calculate the sample percentage, divide the number of lies identified by the lie detector machine by the total number of lies in the sample, and then multiply by 100 to get the percentage.

Sample percentage = (Number of lies identified / Total number of lies) * 100 Sample percentage = (31 / 48) * 100 Sample percentage ≈ 64.5833%

Now, let's construct a 99% confidence interval estimate of the population percentage of lies caught by the lie detector machine.

The formula for the confidence interval for a proportion (percentage) is:

Confidence Interval = Sample percentage ± Margin of error

To calculate the margin of error, we first need to find the standard error (SE) of the sample proportion:

SE = √[(Sample percentage * (1 - Sample percentage)) / Sample size]

Sample size = 48 (total number of lies in the sample) Sample percentage = 31/48 ≈ 0.6458333

SE = √[(0.6458333 * (1 - 0.6458333)) / 48] SE ≈ 0.0658

Next, we need to find the critical value corresponding to a 99% confidence level. Since the sample size is relatively large (48 lies in the sample), we can use the z-table. The critical z-value for a 99% confidence level is approximately 2.576.

Now, calculate the margin of error:

Margin of error = Critical value * Standard error Margin of error = 2.576 * 0.0658 ≈ 0.1695

Finally, construct the confidence interval:

Confidence Interval = Sample percentage ± Margin of error Confidence Interval = 64.5833% ± 0.1695

The 99% confidence interval estimate of the population percentage of lies caught by the lie detector machine is approximately 64.5833% ± 0.1695, which means we are 99% confident that the true population percentage of lies caught by the machine lies within this range.