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A study of MBA graduates by University of Oregon Survey $1999$ revealed that MBA graduates have several expectations of prospective employers beyond their base pay. In particular, according tro the study $46\%$ expect a performance-based related bonus, $46\%$ expect stock options, $42\%$ expect a signing bonus, $28\%$ expect extra vacation, $25\%$ expect tuition reimbursement, $24\%$ expect health insurance and $19\%$ expect guaranteed annual bonuses. Suppose a study is conducted in an ensuing year to see whether these expectations have changed.

If $125$ MBa graduates are randomly selected and if $66$ expect stock options, does this result provide enough evidence to declare that a significantly higher proportion of MBAs expect stock options? Let $\alpha = 0.05$.

If the proportion really is $0.50$, what is the probability of committing a Type II error?

What is the probability of committing a Type II error if the figure is really $0.55$? $0.60$?

Thank you!!

PS - I am thinking of using the Z distribution but I am not sure where to use the 46% who expect stock options. Help!

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Are you certain this question is related to the Riemann Hypothesis? –  Asal Beag Dubh Mar 19 '13 at 18:42
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Best tagging error ever. –  Erick Wong Mar 19 '13 at 18:47
    
Oooppsss.... sorry :( –  user59117 Mar 19 '13 at 18:49
    
@user59117: don't worry, I was just making a little joke. :) –  Asal Beag Dubh Mar 19 '13 at 19:10
    
:) Thanks and let me know if you can help! –  user59117 Mar 19 '13 at 19:36
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1 Answer

66/125 = 0.528

Your null hypothesis is that the ratio p = 0.46

Under your null hypotehsis, the data should be distributed as

Binomial (125, 0.46). The variance is $125 \times 0.46 \times (1-0.46) = 31.05$ The standard deviation $\sigma$ is 5.57.

We here appeal to the central limit thereom, using the formula you should be familiar with $(z-\mu)\sqrt{(n)}/\sigma$

$(0.528-0.46)\times\sqrt{125}/5.57$ = ....

Then you check this against your Z table to check whether you accept or reject the null hypothesis.

Do you understand what type II error is, and know how to compute it in the case when the distribution you are testing is normal?

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