Question

the time to complete an exam is approximately normal with a mean $mu = 55$ minutes and a standard deviation $sigma = 3$ minutes. the bell curve below represents the distribution for testing times. the scale on the horizontal axis is equal to the standard deviation. 1. fill in the indicated boxes. 2. used the empirical rule to complete the following statements: 95% of testing times were between minutes and minutes. % of the testing times were between 52 minutes and 58 minutes.

Explanation:

Step1: Calculate values for bell - curve boxes

For a normal distribution with mean \(\mu = 55\) and standard deviation \(\sigma=3\):

• \(\mu - 3\sigma=55 - 3\times3=55 - 9 = 46\)
• \(\mu - 2\sigma=55-2\times3 = 55 - 6=49\)
• \(\mu-\sigma=55 - 3=52\)
• \(\mu=55\)
• \(\mu+\sigma=55 + 3=58\)
• \(\mu + 2\sigma=55+2\times3=55 + 6 = 61\)
• \(\mu+3\sigma=55+3\times3=55 + 9=64\)

Step2: Use the Empirical Rule for 95% interval

The Empirical Rule states that for a normal distribution, approximately 95% of the data lies within \(\mu\pm2\sigma\).
\(\mu - 2\sigma=55-2\times3 = 49\) and \(\mu + 2\sigma=55 + 2\times3=61\)

Step3: Use the Empirical Rule for 52 - 58 interval

Since \(52=\mu-\sigma\) and \(58=\mu+\sigma\), by the Empirical Rule, approximately 68% of the data lies within \(\mu\pm\sigma\)

Answer:

1. The values for the bell - curve boxes (from left to right) are \(46\), \(49\), \(52\), \(55\), \(58\), \(61\), \(64\)
2. 95% of testing times were between \(49\) minutes and \(61\) minutes. \(68\%\) of the testing times were between \(52\) minutes and \(58\) minutes.