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Significance of difference in proportions

What I'm looking at immediately is "Using the .05 cutoff figure, find an appropriate critical region for testing the null hypothesis 'theta is greater than or equal to .154 against the alternate hypothesis 'theta is less than .154' and provide a Neyman-Pearson analysis.".

Background facts for the problem:

In the county, 15.4 % of certified teachers were African-American. Of the last 405 hires, the school district had hired only 15 African-American teachers. In a discrimination suit, the plaintiff argued that this was unlikely to happen by chance. Thus the significance testing.

Background information on how I need the solution:

I'm REALLY looking for a step-by-step explanation of how to do this, so that I can turn around and do a number of similar problems on my own. It's a multi-part problem and it's possible that I am stuck now because I have done one of the earlier parts incorrectly. In light of this possibility, I have attached my solutions to the earlier parts of the problem (as Excell files). The earlier parts of the problem are below:

Answered on sheet 1 of file "QM 8 1a": "Compute the P-value for the data and provide a P-value analysis." (There's a second column of math for a different population, please ignore it.) There are some related graphs in charts 1-4.

Answered on sheet 1 of file "QM 8 1b": "Produce a graph of the likelihood function and provide a law of likelihood analysis." (Again, ignore the second column.)

Answered on sheet 3 of file "QM 8 1a": Using the .05 and .025 cutoff figures, find an appropriate rejection region for the null hypothesis 'theta is greater than or equal to .154' and provide a significance test analysis. Do the same for 'theta is greater than or equal to .057.'."

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Solution Preview

What I'm looking at immediately is "Using the .05 cutoff figure, find an appropriate critical region for testing the null hypothesis theta is greater than or equal to .154 against the alternate hypothesis 'theta is less than .154' and provide a Neyman-Pearson analysis.".

Background facts for the problem:

In the county, 15.4 % of certified teachers were African-American. Of the last 405 hires, the school district had hired only 15 African-American teachers. In a discrimination suit, the plaintiff argued that this was unlikely to happen by chance. Thus the significance testing.

The cutoff figure is 0.05
This means that the level of significance for testing of hypothesis is α=0.05

Null hypothesis is H0:p=0.154. 15.4% of certified teachers are African-American
Alternative hypothesis is H1:p is not equal to 0.154. The proportion of African-American certified teachers is not equal to 15.4%

Population proportion of success p= 15.40%
Therefore ...

Solution Summary

The solution tests the significance of difference in proportions.

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