Question

A market surveyor wishes to know how many energy drinks teenagers drink each week. They want to construct a 85% confidence interval for the mean and are assuming that the population standard deviation for the number of energy drinks consumed each week is 1.2. The study found that for a sample of 830 teenagers the mean number of energy drinks consumed per week is 7.3. Construct the desired confidence interval. Round your answers to one decimal place.

Answer 1

Answer:

$7.3-1.440\frac{1.2}{\sqrt{830}}=7.240$    

$7.3+1.440\frac{1.2}{\sqrt{830}}=7.360$    

So on this case the 85% confidence interval would be given by (7.2;7.4)  

Step-by-step explanation:

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".

$\bar X=7.3$ represent the sample mean

$\mu$ population mean (variable of interest)

$\sigma =1.2$ represent the population standard deviation

n=830 represent the sample size  

Solution to the problem

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

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

Since the Confidence is 0.85 or 85%, the value of $\alpha=0.15$ and $\alpha/2 =0.075$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.075,0,1)".And we see that $z_{\alpha/2}=1.440$

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

$7.3-1.440\frac{1.2}{\sqrt{830}}=7.240$    

$7.3+1.440\frac{1.2}{\sqrt{830}}=7.360$    

So on this case the 85% confidence interval would be given by (7.2;7.4)