Answer to Question #93415 in Statistics and Probability for gibby

Question

The time taken to pluck the tea leaves in a certain plantation has a normal distribution with a mean of 25 hours per week and a standard deviation of 4 hours. Calculate the probability that the tea leaves can be plucked at this plantation in the following period of time:



More than 30 hours
Between 18 and 34 hours
Between 25 and 34 hours

Answer 1

Let $Z=\frac{t-\mu}{\sigma}$ where $t$ is period of time $\mu$ is mean value of t and $\sigma$ is standard deviation of t.

Z has standard normal distribution

$P(a<t\leq b)=P(\frac{a-\mu}{\sigma}<\frac{t-\mu}{\sigma}\leq\frac{b-\mu}{\sigma})=\\ P(\frac{a-\mu}{\sigma}<Z\leq\frac{b-\mu}{\sigma})=P(\frac{a-25}{4}<Z\leq\frac{b-25}{4})=$

$P(Z\leq\frac{b-25}{4})-P(Z\leq\frac{a-25}{4})$

1)

$P(t>30)=1-P(Z\leq\frac{30-25}{4})=\\ 1-P(Z\leq\frac{5}{4})=1-0.8944=0.1056$

2)

$P(18<t<34)=\\ P(Z\leq \frac{34-25}{4}=2.25)-P(Z\leq \frac{18-25}{4}=-1.75)=\\ 0.9878-0.0401=0.9477$

3)

$P(25<t<34)=\\ P(Z\leq \frac{34-25}{4}=2.25)-P(Z\leq \frac{25-25}{4}=0)=\\ 0.9878-0.5=0.4878$

Answer: $P(t>30)=0.1056$ , $P(18<t<34)=0.9477$ , $P(25<t<34)=0.4878$