Find the probability that a single randomly selected value is less than 994 dollars.
Find the probability that a sample of size $n=65$ is randomly selected with a mean less than 994 dollars.
Answer:
Given:
The population mean $(μ) = 980$
The population standard deviation $(σ) = 288$
The sample size $(n) = 65$
Solution:
(a) The probability that a single randomly selected value is less than 994 dollars :
$$P(x < 994) = P\left(\frac{x – \mu}{\sigma} < \frac{994 – 980}{288}\right)$$ $$= P(z < 0.049)$$ $$= 0.5195$$
(b) The probability that a sample of size n=65 is randomly selected with a mean less than 994 dollars :
$$P(M < 994) = P\left(\frac{M – \mu}{\frac{\sigma}{\sqrt{n}}} < \frac{994 – 980}{\frac{288}{\sqrt{65}}}\right)$$ $$= P(z < 0.392)$$ $$= 0.6525$$
