Question

The times taken to assemble a clock at a factory are approximately normally distributed with a given mean U= 3 hr and standard deviation Q = 0.5 hr. What percentage of the times are between 2 hr and 4 hr?
A)34%
B)47.5%
C)68%
D)95%

Answer 1

Let \(X\) be the time taken to assemble the clock. X is normally distributed, so you can standardize it using :
\[Z = \frac{X-\mu}{\sigma} \] where \(\mu\) is the mean and \(\sigma\) is the standard deviation.

SO the problem is basically asking for \(P(2 \le X \le 4)\). So you can do this:
\[ P(2 \le X \le 4)=P\left(\frac{2-3}{0.5} \le \frac{X-3}{0.5}\le\frac{4-3}{0.5}\right)=P(-2 \le Z \le 2)\]

Answer 2

Then you could
- Use a standard normal distribution to find this probability.
- Use a calculator/software to compute it (if you are not required to learn to use normal tables)
- Use the empirical rule that says that x% of the data will be within 2 standard deviations of the mean.