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

`7.3-1.440(1.2)/(√(830))=7.240`

`7.3+1.440(1.2)/(√(830))=7.360`

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

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)(s)/(√(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(1.2)/(√(830))=7.240`

`7.3+1.440(1.2)/(√(830))=7.360`

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