We want to be rich In a recent year, 73 % of first-year college students responding to a national survey identified "being very well-off financially" as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important. Is there good evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value, 73 %? Carry out a test at the $\alpha =0.05$ significance level to help answer this question.

The significance level of an occasion (like a statistical test) is the probability that the occasion might have happened by some coincidence. If the level is quite low, that is to say, the probability of occurring by chance is tiny, we say the occasion is significant.

The number of students is n = $200$

population proportion = $73\%=0.73$

In favour of well being x = $132$

$\stackrel{-}{p}=\frac{x}{n}=\frac{132}{200}=0.66$

calculating the null and alternative hypotheses,

$\begin{array}{r}{H}_0:p=0.73\\ H_0:p\ne 0.73\end{array}$

Using,

role="math" localid="1652938282434" $z=\frac{\stackrel{-}{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}=\frac{0.66-0.73}{\sqrt{{\displaystyle \frac{0.73(1-0.73)}{200}}}}=-2.23$

The p-value is $=2\times P(z>|z\left|\right)=0.0258$

The p-value is less than the significance level hence, there is good evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value.