Question

question 4 (1 point)
the scores on the most recent exam for a math course had a mean of 76 and a standard deviation of 9.
95% of the scores were between
and
question 5 (1 point)
the scores on the most recent exam for a math course had a mean of 76 and a standard deviation of 9.
68% of the scores were between
and
question 6 (1 point)
the scores on the most recent exam for a math course had a mean of 76 and a standard deviation of 9.
99.7% of the scores were between
and

Explanation:

Step1: Recall the empirical rule

For a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\sigma\)) of the mean (\(\mu\))
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\sigma\)) of the mean (\(\mu\))

Step2: Calculate for \(95\%\)

Given \(\mu = 76\) and \(\sigma=9\)
The lower bound: \(\mu - 2\sigma=76 - 2\times9=76 - 18 = 58\)
The upper bound: \(\mu + 2\sigma=76+ 2\times9=76 + 18 = 94\)

Step3: Calculate for \(68\%\)

The lower bound: \(\mu-\sigma=76 - 9=67\)
The upper bound: \(\mu+\sigma=76 + 9=85\)

Step4: Calculate for \(99.7\%\)

The lower bound: \(\mu - 3\sigma=76-3\times9=76 - 27 = 49\)
The upper bound: \(\mu+3\sigma=76 + 3\times9=76+27 = 103\)

Answer:

Question 4: \(58\) and \(94\)
Question 5: \(67\) and \(85\)
Question 6: \(49\) and \(103\)