Respuesta :
Answer:
[tex]P(\bar X\leq 4.55)=0.0142[/tex]
c. 0.0142
Step-by-step explanation:
1) Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu=4.59,\sigma=0.1)[/tex]
And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
On this case [tex]\bar X \sim N(4.59,\frac{0.1}{\sqrt{16}})[/tex]
2) Solution to the problem
We want this probability:
[tex]P(\bar X\leq 4.55)[/tex]
The question on this case is ".Find the probability that the average price for 30 gas stations is less than $4.55". So then our value for n=30.
If we apply the formula for the z score to our probability we got this:
[tex]P(\bar X \leq 4.55)=P(Z\leq \frac{4.55-4.59}{\frac{0.1}{\sqrt{30}}})=P(Z<-2.19)[/tex]
And using the normal standard table or the following excel code we calculate the probability "=NORM.DIST(-2.19,0,1,TRUE)"
[tex]P(\bar X\leq 4.55)=0.0142[/tex]
