Question

A soft drink manufacturer wishes to know how many soft drinks teenagers drink each week. They want to construct a 98% confidence interval for the mean and are assuming that the population standard deviation for the number of soft drinks consumed each week is 1.3. The study found that for a sample of 2549 teenagers the mean number of soft drinks consumed per week is 5.4.

Construct the desired confidence interval. Round your answers to one decimal place.

Answer 1

`5.4-2.326(1.3)/(√(2549))=5.340`

`5.4+2.326(1.3)/(√(2549))=5.460`

So on this case the 98% confidence interval would be given by (5.3;5.5) after round

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=5.4` represent the sample mean

`\mu` population mean (variable of interest)

`\sigma=1.3` represent the sample standard deviation

n=2549 represent the sample size

Solution to the problem

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

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

Since the Confidence is 0.98 or 98%, the value of
`\alpha=0.02` and
`\alpha/2 =0.01`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.01,0,1)".And we see that
`z_(\alpha/2)=2.326`

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

`5.4-2.326(1.3)/(√(2549))=5.340`

`5.4+2.326(1.3)/(√(2549))=5.460`

So on this case the 98% confidence interval would be given by (5.3;5.5) after round