the average age of private vehicles in Marilao Bulacan is 6 years. Assume that the standard deviation is 2 years. A random sample of 10 private vehicles is chosen, find the probability that the mean of private vehicles ages is
a. between 3 to 9 years
b. over 6 years
Here, we assume that the average age of private vehicles in Marilao Bulacan is normally distributed.
Let
be a random variable representing the average age of private vehicles in Marilao Bulacan from a sample of size
, then the probability that
is less than a value
is given by
, where
is the population mean and
is the population standard deviation.
Here,
,
, and
.
a.) Thus, the probability that the mean of private vehicles ages is between 3 to 9 years is given by![Rendered by QuickLaTeX.com P(3<\bar{X}<9)=P(\bar{X}<9)-P(\bar{X}<3)=P\left(z<\dfrac{9-6}{2/\sqrt{10}}\right)-P\left(z<\dfrac{3-6}{2/\sqrt{10}}\right)](https://acedstudy.com/wp-content/ql-cache/quicklatex.com-fc311097bfc6ad2f21035a2fbb360e6e_l3.png)
Using the table of areas under the standard normal curve or a calculator, we have
.
Therefore, the probability that the mean of private vehicles ages is between 3 to 9 years is 1.
b.) The probability that the mean of private vehicles ages is over 6 years is given by![Rendered by QuickLaTeX.com P(\bar{X}>1)=1-P(\bar{X}<6)=1-P\left(z<\dfrac{6-6}{2/\sqrt{10}}\right)=1-P(z<0)](https://acedstudy.com/wp-content/ql-cache/quicklatex.com-877bb2a255c9609c5f145a4a052a12a0_l3.png)
Using the table of areas under the standard normal curve or a calculator, we have
.
Therefore, the probability that the mean of private vehicles ages is over 6 years is 0.5.