Question

Scores on the SAT Mathematics test are believed to be normally distributed. The scores of a simple random sample of five students who recently took the exam are 570, 620, 710, 540 and 480. We want to find a 95% confidence interval of the population mean of SAT math scores. Calculate the point estimate.

Answer 1

The mean calculated for this case is
`\bar X=584`

And the 95% confidence interval is given by:

`584-2.776(86.776)/(√(5))=476.271`

`584+2.776(86.776)/(√(5))=691.729`

So on this case the 95% confidence interval would be given by (476.271;691.729)

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` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

Solution to the problem

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 mean and the sample deviation we can use the following formulas:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

`s=\sqrt{(\sum_(i=1)^n (x_i-\bar X))/(n-1)}` (3)

The mean calculated for this case is
`\bar X=584`

The sample deviation calculated
`s=86.776`

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

`df=n-1=5-1=4`

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(0.025,4)".And we see that
`t_(\alpha/2)=2.776`

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

`584-2.776(86.776)/(√(5))=476.271`

`584+2.776(86.776)/(√(5))=691.729`

So on this case the 95% confidence interval would be given by (476.271;691.729)