Question

a decade - old study found that the proportion, p, of high school seniors who believed that \getting rich\ was an important personal goal was 70%. a researcher decides to test whether or not that percentage still stands. he finds that, among the 205 high school seniors in his random sample, 147 believe that \getting rich\ is an important goal. can he conclude, at the 0.10 level of significance, that the proportion has indeed changed?
perform a two - tailed test. then complete the parts below.
carry your intermediate computations to three or more decimal places. (if necessary, consult a list of formulas.)
(a) state the null hypothesis ( h_0 ) and the alternative hypothesis ( h_1 ).
( h_0 ):
( h_1 ):
(b) determine the type of test statistic to use.
(c) find the value of the test statistic. (round to three or more decimal places.)
(d) find the p - value. (round to three or more decimal places.)
(e) can we conclude that the proportion of high school seniors who believe that \getting rich\ is an important goal has changed?

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0\) is that the proportion \(p\) has not changed. So \(H_0:p = 0.7\). The alternative hypothesis \(H_1\) is that the proportion has changed, so \(H_1:p
eq0.7\)

Step2: Determine the test - statistic

We use the \(z\) - test statistic for a proportion. The formula for the \(z\) - test statistic is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(\hat{p}=\frac{x}{n}\), \(x = 147\) and \(n=205\)
First, calculate \(\hat{p}\): \(\hat{p}=\frac{147}{205}\approx0.717\)
Then, substitute into the \(z\) - formula:

$$
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$$

Step3: Calculate the p - value

Since it is a two - tailed test, the \(p\) - value is \(2\times(1 - P(Z<|z|))\). Using a standard normal table or calculator, \(P(Z < 0.531)\approx0.702\)
So the \(p\) - value \(=2\times(1 - 0.702)=2\times0.298 = 0.596\)

Step4: Make a decision

Since the \(p\) - value \(=0.596>0.10\) (the level of significance \(\alpha = 0.10\)), we fail to reject the null hypothesis.

Answer:

(a) \(H_0:p = 0.7\), \(H_1:p
eq0.7\)
(b) \(z\) - test statistic
(c) \(z\approx0.531\)
(d) \(p - value\approx0.596\)
(e) No, we cannot conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed.