Question

The article "Americans Seek Spiritual Guidance on Web" (San Luis Obispo Tribune, October 12,2002 ) reported that \(68 \%\) of the general population belong to a religious community. In a survey on Internet use, \(84 \%\) of "religion surfers" (defined as those who seek spiritual help online or who have used the web to search for prayer and devotional resources) belong to a religious community. Suppose that this result was based on a sample of 512 religion surfers. Is there convincing evidence that the proportion of religion surfers who belong to a religious community is different from \(.68\), the proportion for the general population? Use \(\alpha=.05\).

Short answer

To provide a short answer, the actual computation of the z score and P-value are necessary. After that, based on the comparison of the P-value to \(\alpha = 0.05\), we would know whether to reject or fail to reject the null hypothesis, and hence could conclude if there is convincing evidence that the proportion of religion surfers who belong to a religious community is different from \(0.68\), as that the proportion for the general population.

Step-by-step solution

State the Hypotheses

We start by stating the null and alternative hypotheses. The null hypothesis assumes no difference between the two proportions, while the alternative hypothesis states otherwise. Let \( p \) denote the proportion of 'religion surfers' who belong to a religious community.\n\nNull hypothesis \( H_0 \): \( p = 0.68 \)\nAlternative hypothesis \( H_a \): \( p ≠ 0.68 \)

Calculate the Test Statistic

We'll use the formula for the two-proportion z-test to find the z score:\n\n\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]\n\nwhere:\n\(\hat{p}\) is the sample proportion (0.84),\n\(p_0\) is the stated population proportion (0.68), and\n\(n\) is the sample size (512).\n\nCarrying out the calculation yields: \n\n\[ z = \frac{0.84 - 0.68}{\sqrt{\frac{0.68(1 - 0.68)}{512}}} \]

Find the P-value

Once we find the z score, we can determine the P-value, which is the probability under the null hypothesis of obtaining a z score as extreme as the one calculated. The P-value is found using a z-table or statistical software.

Draw Conclusion

We compare the P-value to the significance level, \(\alpha = 0.05\). \n\nIf the P-value is less than \(\alpha\), we reject the null hypothesis and conclude that the proportion of religion surfers belonging to a religious community is significantly different from that of the general population.\n\nOtherwise, if the P-value is greater than \(\alpha\), we fail to reject the null hypothesis, implying that the proportion for religion surfers may be equal to 0.68.