Question

How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and money won by those winners is listed below. Construct a 95% confidence interval to estimate the mean winnings for all players on Jeopardy.$26,650 $6,060 $52,820 $8,490 $13,660$25,840 $49,840 $23,790 $51,480 $18,960$990 $11,450 $41,810 $21,060 $7,860

Answer 1

The 95% confidence interval would be given by (14444.04;33657.30)

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".

Data: $26,650 $6,060 $52,820 $8,490 $13,660$25,840 $49,840 $23,790 $51,480 $18,960$990 $11,450 $41,810 $21,060 $7,860

We can calculate the mean and the deviation from these data with the following formulas:

`\bar X= (\sum_(i=1)^n x_i)/(n)`

`s=\sqrt{(\sum_(i=1)^n (x_i -\bar X)^2)/(n-1)}`

`\bar X=24050.67` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=17386.13 represent the sample standard deviation

n=15 represent the sample size

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=15-1=14`

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 tabel to find the critical value. The excel command would be: "=-T.INV(0.025,14)".And we see that
`t_(\alpha/2)=2.14`

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

`24050.67-2.14(17386.13)/(√(15))=14444.04`

`24050.67+2.14(17386.13)/(√(15))=33657.30`

So on this case the 95% confidence interval would be given by (14444.04;33657.30)