Question

An article in an educational journal discussing a non-conventional high school reading program reports, with 95% confidence, the average score on the reading portion of the SAT is between 520 and 560. The study was based on a random sample of 20 students and it was assumed that SAT scores would be approximately normally distributed. Determine the average SAT reading score of the 20 students sampled.

Answer 1

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

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

For this case the confidence interval at 95% of confidence calculated was (520,560)

And from this case we can estimate the average with the following formula:

`\bar X = (Lower+Upper)/(2)`

Since the average represent the midpoint of the interval, if we replace we got:

`\bar X = (520+560)/(2)= 540`

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=20 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)

For this case the confidence interval at 95% of confidence calculated was (520,560)

And from this case we can estimate the average with the following formula:

`\bar X = (Lower+Upper)/(2)`

Since the average represent the midpoint of the interval, if we replace we got:

`\bar X = (520+560)/(2)= 540`