For the culminating project in this course, you will be asked to utilize statist
ID: 3292904 • Letter: F
Question
For the culminating project in this course, you will be asked to utilize statistical concepts in a real-world context. In particular, you will gather real-world data and employ one of the following techniques:
Calculate a confidence interval for one population mean
Run a hypothesis test for one population mean, with standard deviation known
Run a hypothesis test for one population mean, with standard deviation unknown
In each situation, you will be graded on the following:
Discussion of the context
Data collection methods
Data analysis
Discussion of significance of the results
Explanation / Answer
Calculate a confidence interval for one population mean
A confidence interval shows how confident are we given the value of the parameter lies in that particular range. A confidence interval provides an “interval estimate” of an unknown parameter (as opposed to a “point estimate”). It is designed to contain the parameter’s value with some stated probability. The width of the interval provides a measure of the precision accuracy of the estimator involved. A 100(1-a )% confidence interval for q is defined by specifying random variables Q1 and Q2 such that P(Q1 <q <Q2) = 1-a. Rightly or wrongly, a = 0.05 leading to a 95% confidence interval, is by far the most common case used in practice and we will tend to use this in most of our calculations. Confidence intervals are not unique. In general they should be obtained via the sampling distribution of a good estimator, in particular the maximum likelihood estimator. Even then there is a choice between one-sided and two-sided intervals and between equal-tailed and shortest-length intervals although these are often the same, eg for sampling distributions that are symmetrical about the unknown value of the parameter.
Run a hypothesis test for one population mean, with standard deviation known
For one population mean, in sampling from a N(µ, 2) distribution with known value of 2, a pivotal quantity is: Z= (X-µ)/(/n) which is N(0,1).
Run a hypothesis test for one population mean, with standard deviation unknown
When we deal confidence intervals for a normal mean µ in the case where the standard deviation 2 was known we use the above case. In practice this is unlikely to be the case and so we need a different pivotal quantity for the realistic case when 2is unknown.
Fortunately there is a similar pivotal quantity readily available and that is the t-result:
T= (X-µ)/(S/n)~tn-1
Here S is the estimated standard deviation(obtained by taking the square root of the value obtained by dividing the sum of the square of the differences of the observations from their mean by (n-1). This statistic follows the t distribution.
Standard method of hypothesis testing is then applied to the sample values obtained from the survey which employs the correct data collection method and the above statistics are calculated and inferences are drawn based on the result.