Question

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants toestimate its average number of unoccupied seats per flight over the past year. 225 flight records are randomly selected and the number of unoccupied seats is noted with a sample mean of 11.6 seats and a standard deviation of 4.1 seats. Calculate a 90% confidence interval for , the mean number of unoccupied seats per flight during the past year. A. [4.86 18.34]B. [11.25 11.95]C. [11.57 11.63]D.[11.15 12.05]E. [11.30 12.20]

Answer 1

`11.6-1.65(4.1)/(√(225))=11.149`

`11.6+1.65(4.1)/(√(225))=12.051`

So on this case the 90% confidence interval would be given by (11.15;12.05) And the best option is:

D.[11.15 12.05]

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=11.6` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=4.1 represent the sample standard deviation

n=225 represent the sample size

2) Confidence interval

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

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

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=225-1=224`

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 tabel to find the critical value. The excel command would be: "=-T.INV(0.05,224)".And we see that
`t_(\alpha/2)=1.65`

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

`11.6-1.65(4.1)/(√(225))=11.149`

`11.6+1.65(4.1)/(√(225))=12.051`

So on this case the 90% confidence interval would be given by (11.15;12.05) And the best option is:

D.[11.15 12.05]