Question

what is the area of the shaded region under the normal curve? 1 0.95 0.997 0.68

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(\mu\pm\sigma\)
• Approximately \(95\%\) of the data lies within \(\mu\pm2\sigma\)
• Approximately \(99.7\%\) of the data lies within \(\mu\pm3\sigma\)

Step2: Interpret the shaded region

The shaded region represents the area from \(-\infty\) to \(\mu + 3\sigma\). Since the total area under a normal curve is \(1\), and the area within \(\mu\pm3\sigma\) is \(0.997\) (for the symmetric part around \(\mu\)), but when we consider from \(-\infty\) to \(\mu + 3\sigma\), we note that the area to the left of \(\mu\) is \(0.5\) and the area from \(\mu\) to \(\mu + 3\sigma\) is \(\frac{0.997}{2}=0.4985\). The sum \(0.5 + 0.4985\approx0.997\)

Answer:

\(0.997\)