Question

the testing times for a group of college students were normally distributed with a mean of \\( \mu = 35 \\) minutes and a standard deviation of \\( \sigma = 2.1 \\) minutes.
the bell curve below represents the distribution for testing times. the scale on the horizontal axis is equal to the standard deviation.
fill in the indicated boxes.
bell curve image with \\( \mu = 35 \\), \\( \sigma = 2.1 \\), and horizontal axis labels \\( \mu - 3\sigma \\), \\( \mu - 2\sigma \\), \\( \mu - \sigma \\), \\( \mu \\), \\( \mu + \sigma \\), \\( \mu + 2\sigma \\), \\( \mu + 3\sigma \\) and empty boxes below
used the empirical rule to complete the following statements:
68% of testing times were between \\( \square \\) minutes and \\( \square \\) minutes.
95% of testing times were between \\( \square \\) minutes and \\( \square \\) minutes.
99.7% of testing times were between \\( \square \\) minutes and \\( \square \\) minutes.
50% of testing times were below \\( \square \\) minutes.
question help: \\( \square \\) video \\( \square \\) message instructor

Explanation:

Step1: Calculate values for Empirical Rule (68 - 95 - 99.7)

The Empirical Rule states that for a normal distribution:

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

Given \( \mu = 35\) and \( \sigma=2.1\)

For 68%:
\( \mu-\sigma=35 - 2.1=32.9\)
\( \mu+\sigma=35 + 2.1 = 37.1\)

For 95%:
\( \mu - 2\sigma=35-2\times2.1=35 - 4.2 = 30.8\)
\( \mu+2\sigma=35 + 2\times2.1=35+4.2 = 39.2\)

For 99.7%:
\( \mu-3\sigma=35-3\times2.1=35 - 6.3=28.7\)
\( \mu + 3\sigma=35+3\times2.1=35 + 6.3=41.3\)

For the 50% statement: In a normal distribution, the mean \( \mu\) divides the data into two equal - halves. So 50% of the data is below the mean.

Answer:

• 68% of testing times were between \(32.9\) minutes and \(37.1\) minutes.
• 95% of testing times were between \(30.8\) minutes and \(39.2\) minutes.
• 99.7% of testing times were between \(28.7\) minutes and \(41.3\) minutes.
• 50% of testing times were below \(35\) minutes.