Question

The fuel economy of a 2011 Lexus RX 350 2WD 6 cylinder 3.5 L automatic 5-speed using premium fuel is a normally distributed random variable with a mean of μ = 25.0 MPG and a standard deviation of σ = 1.25 MPG. (a) What is the standard error of X⎯⎯⎯ , the mean from a random sample of 16 fill-ups by one driver? (Round your answer to 4 decimal places.) Standard error of X⎯⎯⎯ (b) Within what interval would you expect the sample mean to fall, with 90 percent probability? (Round your answers to 4 decimal places.) The interval is from to

Answer 1

Answer:

a) $\sigma_{\bar X}= \frac{1.25}{\sqrt{16}}= 0.3125$

b) $25-1.64\frac{1.25}{\sqrt{16}}=24.4875$  

$25+1.64\frac{1.25}{\sqrt{16}}=25.5125$  

Step-by-step explanation:

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".  

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

$X \sim N(25,1.25)$  

Where $\mu=25$ and $\sigma=1.25$

Part a

We know that the distribution for the sample mean $\bar X$ is given by:

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

The standard error is given by:

$\sigma_{\bar X}= \frac{\sigma}{\sqrt{n}}$

And replacing we got:

$\sigma_{\bar X}= \frac{1.25}{\sqrt{16}}= 0.3125$

Part b

The confidence interval for the mean is given by the following formula:  

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (1)  

Since the Confidence is 0.90 or 90%, the value of $\alpha=0.1$ and $\alpha/2 =0.05$, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that $z_{\alpha/2}=1.64$  

Now we have everything in order to replace into formula (1):  

$25-1.64\frac{1.25}{\sqrt{16}}=24.4875$  

$25+1.64\frac{1.25}{\sqrt{16}}=25.5125$