Answer:
[tex]2.724< \mu <3.076[/tex]
Step-by-step explanation:
1) Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
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".
[tex]\bar X=2.90[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
[tex]\sigma=0.45[/tex] represent the population standard deviation
n=25 represent the sample size
We have the following distribution for the random variable:
[tex]X \sim N(\mu , \sigma=0.45)[/tex]
And by the central theorem we know that the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
2) Confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], 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 [tex]z_{\alpha/2}=\pm 1.96[/tex]
Now we have everything in order to replace into formula (1):
[tex]2.90-1.96\frac{0.45}{\sqrt{25}}=2.724[/tex]
[tex]2.90+1.96\frac{0.45}{\sqrt{25}}=3.076[/tex]
So on this case the 95% confidence interval would be given by (2.724;3.076)
[tex]2.724< \mu <3.076[/tex]
