Question

A Common Measure of Student Achievement, The National Assessment of Educational Progress (NAEP) is the only assessment that measures what U.S. students know and can do in various subjects across the nation, states, and in some urban districts. Also known as The Nation's Report Card, NAEP has provided important information about how students are performing academically since 1969. NAEP is a congressionally mandated project administered by the National Center for Education Statistics (NCES) within the U.S. Department of Education and the Institute of Education Sciences (IES). NAEP is given to a representative sample of students across the country. Results are reported for groups of students with similar characteristics (e.g., gender, race and ethnicity, school location), not individual students. National results are available for all subjects assessed by NAEP. In the most recent year, The NAEP sample of 1077 young women had mean quantitative score of 275. Individual NAEP scores have a Normal distribution with standard deviation of 60. (This is indicating that the population standard deviation σσ is 60) Find a 99% Confidence Interval for the mean quantitative scores for young women. a) Check that the normality assumptions are met. ? b) What is the 99% confidence interval for the mean quantitative scores for young women? ____ ≤μ≤ _____ c) Interpret the confidence interval obtained in previous question

Answer 1

a) By the central limit theorem we know that if the random variable X="quantitative scores for young women", and we know that this distribution is normal.

`X \sim N(\mu, \sigma=60)`

And we are interested on the distribution for the sample mean
`\bar X`, we know that distribution for the sample mean is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And with that we have the assumptions required to apply the normal distribution to create the confidence interval since the distribution for
`\bar X` is normal.

b)
`270.283\leq \mu \leq 279.717`

c) We are confident (99%) that the true mean for the quantitative scores for young women is between (270.283;3279.717)

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=275` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=60` represent the population standard deviation

n=1077 represent the sample size

Part a

By the central limit theorem we know that if the random variable X="quantitative scores for young women", and we know that this distribution is normal.

`X \sim N(\mu, \sigma=60)`

And we are interested on the distribution for the sample mean
`\bar X`, we know that distribution for the sample mean is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

And with that we have the assumptions required to apply the normal distribution to create the confidence interval since the distribution for
`\bar X` is normal.

Part b

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

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

Since the Confidence is 0.99 or 99%, the value of
`\alpha=0.01` and
`\alpha/2 =0.005`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that
`z_(\alpha/2)=2.58`

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

`275-2.58(60)/(√(1077))=270.283`

`275+2.58(60)/(√(1077))=279.717`

So on this case the 99% confidence interval would be given by (270.283;3279.717)

`270.283\leq \mu \leq 279.717`

Part c

We are confident (99%) that the true mean for the quantitative scores for young women is between (270.283;3279.717)