Answer

High School Longitudinal Study dataset: Confidence Interval

In estimation, confidence intervals are integral metrics that give the range of values that could truly estimate the population statistic from a sample as it helps in testing how well a sample represents a population. . While statistic gives an exact estimation, confidence interval is an ideal range of estimation while taking into account the reality of sampling errors. In this paper is discussed the confidence interval of the mean of ‘X1SES’ variable in the (T1 Socio-economic status composite) in the High School Longitudinal Study dataset.

Confidence Interval of X1SES Variable

The distribution of the variable is presented as in the histogram and boxplots below.

Fig 1: Histogram of the Socio-economic composite Variable. Based on this plot, the variable is approximately normally distributed, setting the right precedence for estimation of the population mean of socio-economic composite.

Fig 2: The boxplot for the Socio-economic composite Variable clearly showing significant outliers. The figure shows the mean as being in the middle (normally distributed) but the effects of outliers on the mean as a measure of central tendency belie its accuracy in estimating the population mean as a point estimate. 

The confidence interval output is as follows.

The 95% confidence interval of the variable is [0.0436, 0.0645]. This means that there is a 95% probability that the population’s mean Socio-economic status composite will always range between 0.0436 and 0.0645.

The 90% confidence interval, however, presents a narrower interval width as shown below.

 There is a 90% confidence that the population’s mean Socio-economic status composite will always range from 0.0453 to 0.0629.

Though the two confidence levels present different margins due to the shifts in size of the margin of error, the higher confidence presents wider margin, which means that wider margins are more appropriate for estimation.

As the use of confidence interval beats point estimation in estimating population parameters in terms of accuracy and risk aversion to being more ideal, it sets up a more realistic estimation measurement as it deals with potential variation due to errors.

The confidence interval can also be used to test hypothesis about the population mean in regards to socio-economic status composite, with significance being that the test statistic falls within the interval.