Question

a set of data has a normal distribution with a mean of 5.1 and a standard deviation of 0.9. sketch a normal curve for the distribution and find the percent of data within each interval.

1. between 6.0 and 6.9
2. greater than 6.9
3. between 4.2 and 6.0
4. less than 4.2
5. less than 5.1
6. between 4.2 and 5.1

test scores are normally distributed with a mean of 76 and a standard deviation of 10.

1. in a group of 230 tests, how many students score above 96.
2. in a group of 230 tests, how many students score below 66.
3. in a group of 230 tests, how many students score within one standard deviation of the mean?

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
For \(x = 96\), \(\mu=76\), \(\sigma = 10\), then \(z=\frac{96 - 76}{10}=\frac{20}{10}=2\)

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(z=\pm1\)
• Approximately \(95\%\) of the data lies within \(z=\pm2\)
• Approximately \(99.7\%\) of the data lies within \(z=\pm3\)

The percentage of data above \(z = 2\) is \(\frac{100\%-95\%}{2}=2.5\%\)

Step3: Calculate the number of students

If there are \(n = 230\) tests, the number of students with scores above \(96\) is \(0.025\times230 = 5.75\approx6\)

Answer:

Approximately \(6\) students score above \(96\)