In a certain survey, 521 people chose to respond to this question: "Should passwords be replaced with biometric security (fingerprints, etc)?" Among the respondents, $$ 51 \% $$ said "yes." We want to test the claim that more than half of the population believes that passwords should be replaced with biometric security. Complete parts (a) through (d) below. a. Are any of the three requirements violated? Can a test about a population proportion using the normal approximation method be used? A. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so the method can be used. B. The conditions $$ \mathrm{np} \geq 5 $$ and $$ \mathrm{nq} \geq 5 $$ are not satisfied, so a test about a population proportion using the normal approximation method cannot be used. C. One of the conditions for a binomial distribution are not satisfied, so a test about a population proportion using the normal approximating method cannot be used. D. The sample observations are not a random sample, so a test about a population proportion using the normal approximating method cannot be used.

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UpStudy AI · Accepted Answer

**Step 1: Define the Hypotheses**

We are testing the claim that more than half of the population supports replacing passwords with biometric security. In terms of proportions, we let $$ p $$ be the true proportion of the population in favor. Then, we set up

- Null hypothesis: $$ H_0: p = 0.5 $$
- Alternative hypothesis: $$ H_a: p > 0.5 $$

**Step 2: Check the Conditions for the Normal Approximation**

For tests about a population proportion using the normal approximation, we need to check:

1. **Randomness:** The data should come from a random sample.
2. **Binomial Condition (independence):** The outcomes should be independent (typically if sampling without replacement from a large population).
3. **Normal Approximation Conditions:** We require both $$ np \geq 5 $$ and $$ n(1-p) \geq 5 $$.

Given that:
- Sample size is $$ n = 521 $$.
- Under the null hypothesis $$ p = 0.5 $$, we calculate
  $$
  np = 521 \times 0.5 = 260.5 \quad \text{and} \quad n(1-p) = 521 \times 0.5 = 260.5.
  $$

Both values are much larger than 5, so the conditions for applying the normal approximation are met.

**Step 3: Conclusion for Part (a)**

Since none of the requirements are violated and the conditions $$ np \geq 5 $$ and $$ n(1-p) \geq 5 $$ are clearly satisfied, we can use a test about a population proportion using the normal approximation method.

Thus, the correct answer is:

**A. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so the method can be used.**
All conditions are met, so a normal approximation test can be used.

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