Question

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.â  Suppose a small group of 11 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with Ï = 0.38 gram.

Find an 80% confidence interval for the average weight of Allen's hummingbirds in the study region. What is the margin of error?

Answer 1

`3.15-1.28(0.38)/(√(11))=3.003`

`3.15+ 1.28(0.38)/(√(11))=3.297`

So on this case the 80% confidence interval would be given by (3.003;3.297)

And the margin of error is given by:

`ME = 1.28(0.38)/(√(11))= 0.147`

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=3.15` represent the sample mean

`\mu` population mean (variable of interest)

`\sigma=0.38` represent the population standard deviation

n=11 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 0.80 or 80%, the value of
`\alpha=0.2` and
`\alpha/2 =0.1`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NOR.INV(0.1,0,1)".And we see that
`z_(\alpha/2)=1.28`

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

`3.15-1.28(0.38)/(√(11))=3.003`

`3.15+ 1.28(0.38)/(√(11))=3.297`

So on this case the 80% confidence interval would be given by (3.003;3.297)

And the margin of error is given by:

`ME = 1.28(0.38)/(√(11))= 0.147`