Question

The lives of a type of electric bulbs are normally distributed with a standard deviation = 25.60 hours. A simple random sample of n=16 such electric bulbs is selected. The sample mean is found to be = 325.78 hours. You are required to construct a confidence interval for the population mean with a confidence coefficient 1 - a = .95 The left limit L of this confidence interval is closest to (3)The Right limit R of this confidence interval is closest to (3)The half width of this confidence interval is closest to (3)

Answer 1

a)
`325.78-1.96(25.60)/(√(16))=313.236`

b)
`325.78+1.96(25.60)/(√(16))=338.324`

c)
`Me=z_(\alpha/2)(\sigma)/(√(n))`

`Me=1.96(25.60)/(√(16))=12.544`

Explanation:

1) 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=325.78` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=25.60` represent the population standard deviation

n=16 represent the sample size

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 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
`z_(\alpha/2)=1.96`

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

The left limit L of this confidence interval is closest to (3)

`325.78-1.96(25.60)/(√(16))=313.236`

The Right limit R of this confidence interval is closest to (3)

`325.78+1.96(25.60)/(√(16))=338.324`

So on this case the 95% confidence interval would be given by (313.236;338.324)

The half width of this confidence interval is closest to (3)

We need to take in count that the margin of error is defined as "half the width of the confidence interval".

And the margin of error is given by:

`Me=z_(\alpha/2)(\sigma)/(√(n))`

`Me=1.96(25.60)/(√(16))=12.544`