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

Answer:

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

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (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:

$\bar X = \frac{Lower+Upper}{2}$

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

$\bar X = \frac{520+560}{2}= 540$

Step-by-step 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:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (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:

$\bar X = \frac{Lower+Upper}{2}$

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

$\bar X = \frac{520+560}{2}= 540$