Question

1) Once again, you have been asked to study the mean yearly tuition at private four-year colleges and universities across the United States. You are interested in making some decisions concerning this population parameter. You select a random sample of 50 universities from this population. The mean tuition at these universities is found to equal $18,205. You still have reason to believe that the population standard deviation of the tuition amounts is known to equal $9,000. At both the 5% and 10% levels of significance, is the mean tuition at this population of universities less than $20,000? In your memo, if appropriate, comment upon the effect of the change in the significance level on your decision.

2 You no longer believe that the population standard deviation in the tuition amounts for this population is a known quantity. You therefore will use the standard deviation in the tuition amounts of a representative sample of universities as an estimate of the unknown population standard deviation. You collect yearly tuition data from a random sample of universities once again. This data is shown in appendix one below. Once again, at each of the 5% and 10% levels of significance, is the mean yearly tuition less than $20,000? Again, in your memo, comment upon the effect of the change in the level of significance on your decision, if necessary. Also, compare, at each level of significance, the results of this portion of the problem to those of the previous part. Account for any difference in your decisions at each level of significance between the two parts of the problem. Make this accounting based not only on a mathematical approach, but rather on a conceptual justification.

            Appendix One: (Tuition Amounts)

            $22,000          $25,412          $18,543          $21,010          $32,500

            $13,476          $18,765          $17,689          $12,378          $21,800

            $19,548          $22,348          $17,659          $18,654          $23,409

            $31,329          $14,489          $15,698          $11,389          $19,901

            $25,671          $18,888          $14,490          $24,468          $15,690

            $13,298          $30,000          $12,390          $21,672          $20,037

            $21,876          $19,090          $21,684          $24,347          $18,000

            $12,769          $17,032          $26,876          $18,923          $15,119

            $15,632          $20,000          $21,769          $15,858          $13,607

            $24,879          $17,540          $14,027          $13,908          $12,690

Answer 1

Solution;

• Here, we are to test:
    
  The test statistics, T is of the form:
  

where,

n = number of observations.
= observed sample mean.
= hypothesized population mean.
= known population standard deviation.

We reject the null hypothesis iff where, is the upper percentile of standard normal distribution.
is given to be 18205
is given to be 9000


Now,


It is observed that,  
ie., the null hypothesis is accepted at 5% level of significance.

It is observed that,  
ie., the null hypothesis is rejected at 10% level of significance.


Here, on increasing our level of significance, that is, on making our critical region more wide, the null hypothesis is rejected. However, on doing so, P(Type I error) has increased to a great extent, and this is risky.

• Here, we are to test:
    
  The test statistics, T₁ is of the form:
  

where,

n = number of observations.
= observed sample mean.
= hypothesized population mean.
s = estimated population standard deviation.

, where, t₄₉ is t distribution with 49 degrees of freedom.
We reject the null hypothesis iff where, is the upper percentile of t distribution with (n-1) degrees of freedom.


Now,
t_49,0.05 = 1.677
t_49,0.1 = 1.299

It is observed that,
ie., the null hypothesis is accepted at 5% level of significance.

It is observed that,
ie., the null hypothesis is accepted at 10% level of significance.

At both the level of significance, the null hypothesis is accepted ie., $20,000 is the mean tuition fees.