Question

According to a Washington Post-ABC News poll, 331 of 502 randomly selected U.S. adults said they would not be bothered if the National Security Agency collected records of personal telephone calls. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a 0.01 significance level.

Short answer

Yes, there is sufficient evidence to conclude that majority of U.S. adults would be bothered if the NSA collected records of personal telephone calls as the p-value is less than the level of significance, leading us to reject the null hypothesis.

Step-by-step solution

Identify the Hypotheses

The null hypothesis \((H_0)\) is to assume the status quo, which in this case is that a majority (more than 50% or 0.5) of U.S. adults would not mind. So, \(H_0: p \leq 0.5\). The alternative hypothesis \((H_a)\) represents the conclusion we want to reach, that a majority would mind. So, \(H_a: p > 0.5\).

Calculate the Proportion in the Sample

The proportion of U.S. adults in the sample who do not mind the NSA collects personal calls records is \(p = 331/502 = 0.6594.\)

Compute the Test Statistics

Now, compute the z-score for this. The formula used is \(z = (p - p_0)/(sqrt((p_0*(1-p_0))/n))\), where \(p_0\) is the assumed proportion in the null hypothesis, \(p\) is the sample proportion and \(n\) is the sample size. The value will be \(z = (0.6594 - 0.5)/(sqrt((0.5*(1-0.5))/502)) = 7.8.\)

Calculate the p-value

Refering to the standard z-table or any statistical tool, the p-value associated with the z score of 7.8 is less than 0.01.

Compare the p-value and the Level of Significance

Here, the p-value < 0.01 which is less than our significance level 0.01.

Make Conclusions

Since the p-value is less than the level of significance, we reject the null hypothesis. There is sufficient statistical evidence at the 0.01 level of significance to conclude that a majority of U.S. adults would be bothered if the NSA collected records of personal telephone calls.