Question

In a recent study,1 2006 randomly selected US adults (age 18 or older) were asked to give the number of people in the last six months ‘‘with whom you discussed matters that are important to you." The average number of close confidants was 2.2, with a standard deviation of 1.4. 1Hampton, K., Goulet, L., Rainie, L., and Purcell, K., ‘‘Social Networking Sites and Our Lives," Pew Research Center, pewresearch.org, June 16, 2011. (a) Find the margin of error for this estimate if we want 90% confidence. Round your answer to three decimal places. margin of error = Enter your answer in accordance to the question statement

Answer 1

a)
`ME=1.64 (1.4)/(√(2006))=0.051`

b) The 90% confidence interval would be given by (2.149;2.251)

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=2.2` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=1.4 represent the sample standard deviation

n=2006 represent the sample size

Part a

The margin of error is given by this formula:

`ME=z_(\alpha/2)(s)/(√(n))`

Since the Confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that
`z_(\alpha/2)=1.64`

`ME=1.64 (1.4)/(√(2006))=0.0512`

Part b

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

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

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

`2.2 - 1.64(1.4)/(√(2006))=2.149`

`2.2 +1.64(1.4)/(√(2006))=2.251`

So on this case the 90% confidence interval would be given by (2.149;2.251)