Question

The probability that a particular student passes on an exam is found to be 0.6. A sample of 20 students are put through a training program before the exam and 15 of them pass the exam. Is the result significant on the 10% level?

I attempted the question and know that n=20, P(successes) = 15 (0.75) and Q(failures) = 0.25 I think.. dont know where the 0.6 comes into play.. please help me

Answer 1

Hi,

Probably you need no longer help with this, but I've attempted to solve it.

The 0.6 can be taken as a given. It's the population proportion. You have:

\(p = 0.6\)
\(p_\bar{x} = 0.75\)
\(1 - p_\bar{x} = 0.25\)
\(n (sample\ size) = 20\)

You're asked to find if 0.75 is a significant result that allows you to reject the hypothesis that the given population proportion is 0.6. Whether it's greater than 0.6 or equal to it.

\(H_0: p = 0.6\)
\(H_1: p > 0.6\)

You are not given the standar deviation, but you can estimate it (SE, Standard Error):

\(SE = \frac{\sqrt{n(p_\bar{x})(1 - p_\bar{x})}}{n} = \frac{\sqrt{20(0.75)(0.25)}}{20} = 0.0968\).

Since the standard deviation is estimated, we must compute a t-statistics rather than a z-statistics:

\(t = \frac{0.75-0.60}{0.0968} = 1.55_{\sigma_{p}}\)

You have 20 - 1 = 19 degrees of freedom an so the t-value is 0.0688. Since this is < than 0.10 you can reject the null hypothesis.