Answer to Question #261466 in Statistics and Probability for Joyce

Question

The engine made by alpha-x for boats have an average power of 220 horsepower and a standard deviation of 15HP. You can assume the distribution of power follows a normal distribution. Beta-y is testing the engines and will dispute the company's claim if the sample mean is less than 215HP. If they take a sample of 4 engines; what is the probability the mean is less than 215? If beta-y samples 100 engines, what is the probability that the mean will be less than 215?

Answer 1

Solution:

We want to find $P(\bar X<215)$.

Since the population follows a normal distribution, we can conclude that $\bar X$  has a normal distribution with mean 220 HP (μ=220) and a standard deviation of $\dfrac σ{\sqrt n}=\dfrac{15}{\sqrt4}=7.5$ HP.

Now,

$P(\bar X<215)=P(z<\dfrac{215-220}{7.5}) \\=P(z<-0.67) \\\approx 0.2514$

Next, the sampling distribution of the sample mean is Normal with mean μ=220

 and standard deviation $\dfrac σ{\sqrt n}=\dfrac{15}{\sqrt{100}}=1.5$

Now,

$P(\bar X<215)=P(z<\dfrac{215-220}{1.5}) \\=P(z<-3.333..) \\\approx 0.00043$