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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To determine if you can use the normal approximation for a test about the population proportion, you need to check the conditions: the sample size \( n \) must be large enough so that both \( np \) and \( nq \) are at least 5. Here, \( p = 0.51 \) and \( q = 1 - p = 0.49 \). Calculating these: - \( np = 521 \times 0.51 \approx 266.71 \) - \( nq = 521 \times 0.49 \approx 254.29 \) Both are greater than 5, so the requirements are satisfied. 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.
