Question

We draw a random sample of size 36 from a population with standard deviation 3.5. If the sample mean is 27, what is a 95% confidence interval for the population mean?

Answer 1

Answer:

The 95% confidence interval for the population mean is between 25.86 and 28.14.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1-0.95}{2} = 0.025$

Now, we have to find z in the Ztable as such z has a pvalue of $1-\alpha$.

So it is z with a pvalue of $1-0.025 = 0.975$, so $z = 1.96$

Now, find the margin of error M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

$M = 1.96\frac{3.5}{\sqrt{36}} = 1.14$

The lower end of the interval is the sample mean subtracted by M. So it is 27 - 1.14 = 25.86

The upper end of the interval is the sample mean added to M. So it is 27 + 1.14 = 28.14

The 95% confidence interval for the population mean is between 25.86 and 28.14.

Answer 2

Answer:

$27-1.96\frac{3.5}{\sqrt{36}}=25.857$  

$27+1.96\frac{3.5}{\sqrt{36}}=28.143$  

The 95% confidence interval would be given by (25.857;28.143). And we can conclude that the true mean with 95% is between (25.857;28.143)

Step-by-step explanation:

Data given

$\bar X=27$ represent the sample mean  

$\mu$ population mean (variable of interest)  

$\sigma=3.5$ represent the population standard deviation  

n=36 represent the sample size  

95% confidence interval  

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 0.95 or 96%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that $z_{\alpha/2}=1.96$  

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

$27-1.96\frac{3.5}{\sqrt{36}}=25.857$  

$27+1.96\frac{3.5}{\sqrt{36}}=28.143$  

The 95% confidence interval would be given by (25.857;28.143). And we can conclude that the true mean with 95% is between (25.857;28.143)