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 number of standard deviations

For problem 15:
The value \(x = 96\), mean \(\mu=76\), standard deviation \(\sigma = 10\).
The \(z - score=\frac{x-\mu}{\sigma}=\frac{96 - 76}{10}=2\)

For problem 16:
The value \(x = 66\), mean \(\mu = 76\), standard deviation \(\sigma=10\).
The \(z - score=\frac{x-\mu}{\sigma}=\frac{66 - 76}{10}=- 1\)

For problem 17:
The interval within one standard deviation of the mean is \(\mu-\sigma\) to \(\mu+\sigma\), i.e., \(76 - 10\) to \(76+10\) or \(66\) to \(86\). The percentage of data within one standard deviation of the mean in a normal distribution is \(68\%\)

Step2: Use the properties of the normal distribution

For problem 15:
The percentage of data above \(z = 2\) is \(2.5\%\) (using the empirical rule: \(95\%\) of data is within \(\mu\pm2\sigma\), so \(5\%\) is outside, half of which is above \(z = 2\))
The number of students \(n=230\times0.025 = 5.75\approx6\)

For problem 16:
The percentage of data below \(z=-1\) is \(16\%\) (using the empirical rule: \(68\%\) of data is within \(\mu\pm\sigma\), so \(32\%\) is outside, half of which is below \(z=-1\))
The number of students \(n = 230\times0.16=36.8\approx37\)

For problem 17:
The number of students \(n=230\times0.68 = 156.4\approx156\)

Answer:

1. Approximately \(6\) students score above \(96\).
2. Approximately \(37\) students score below \(66\).
3. Approximately \(156\) students score within one standard deviation of the mean.