What is the confidence level of each of the following confidence intervals for $\mu$ ? Complete parts a through e. $\newline$ E Click the icon to view the table of normal curve areas. $\newline$ a. $\bar{x}\pm1.96\left(\frac{\sigma}{\sqrt{n}}\right)$ $\newline$ % (Round to two decimal places as needed.)

View step-by-step help

Home

Math Problems

Algebra 2

Solve trigonometric equations II

Full solution

Q. What is the confidence level of each of the following confidence intervals for

\mu

? Complete parts a through e.

\newline

E Click the icon to view the table of normal curve areas.

\newline

\bar{x}\pm1.96\left(\frac{\sigma}{\sqrt{n}}\right)

\newline

% (Round to two decimal places as needed.)

Lookup z-score: The confidence interval given is in the form of a standard normal distribution where the value $1.96$ corresponds to the z-score for a certain confidence level. To find the confidence level, we need to look up the z-score in the standard normal distribution table or use a calculator that provides this functionality.
Find cumulative probability: Using the z-score of $1.96$ , we find the area under the standard normal curve to the left of $z = 1.96$ . This area corresponds to the cumulative probability up to that z-score.
Calculate total area: The area to the left of $z = 1.96$ is approximately $0.975$ . This is because the z-score table or calculator tells us that approximately $97.5\%$ of the data falls to the left of a z-score of $1.96$ in a standard normal distribution.
Determine confidence level: Since the confidence interval is two-tailed (plus and minus the $z$ -score), we need to consider the area in both tails of the distribution. The area in one tail is $1 - 0.975 = 0.025$ . Therefore, the total area in both tails is $2 \times 0.025 = 0.05$ .
Determine confidence level: Since the confidence interval is two-tailed (plus and minus the z-score), we need to consider the area in both tails of the distribution. The area in one tail is $1 - 0.975 = 0.025$ . Therefore, the total area in both tails is $2 \times 0.025 = 0.05$ .To find the confidence level, we subtract the total area in the tails from $1$ . So, the confidence level is $1 - 0.05 = 0.95$ or $95\%$ .