$$ 1 \leftarrow \quad \begin{array}{l} ext { In a certain survey, } 521 ext { people chose to respond to this question: "Should passwords be replaced with biometric security (fingerprints, } \ ext { etc)?" Among the respondents, } 51 \% ext { said "yes." We want to test the claim that more than half of the population believes that passwords } \ ext { should be replaced with biometric security. Complete parts (a) through (d) below. } \ ext { (d) Are any of the three requirements violated? Can a test about a population proportion using the normal approximation method } \ ext { be used? } \ ext { A. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so } \ ext { the method can be used. } \ ext { B. The conditions np } \geq 5 ext { and nq } \geq 5 ext { are not satisfied, so a test about a population proportion using the normal approximation } \ ext { method cannot be used. } \ ext { C. One of the conditions for a binomial distribution are not satisfied, so a test about a population proportion using the normal } \ ext { approximating method cannot be used. } \ ext { D. The sample observations are not a random sample, so a test about a population proportion using the normal approximating } \ ext { method cannot be used. }\end{array} $$ (t)

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { In a certain survey, } 521 	ext { people chose to respond to this question: "Should passwords be replaced with biometric security (fingerprints, } \ 	ext { etc)?" Among the respondents, } 51 \% 	ext { said "yes." We want to test the claim that more than half of the population believes that passwords } \ 	ext { should be replaced with biometric security. Complete parts (a) through (d) below. } \ 	ext { (d) Are any of the three requirements violated? Can a test about a population proportion using the normal approximation method } \ 	ext { be used? } \ 	ext { A. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so } \ 	ext { the method can be used. } \ 	ext { B. The conditions np } \geq 5 	ext { and nq } \geq 5 	ext { are not satisfied, so a test about a population proportion using the normal approximation } \ 	ext { method cannot be used. } \ 	ext { C. One of the conditions for a binomial distribution are not satisfied, so a test about a population proportion using the normal } \ 	ext { approximating method cannot be used. } \ 	ext { D. The sample observations are not a random sample, so a test about a population proportion using the normal approximating } \ 	ext { method cannot be used. }\end{array} $$ (t)

UpStudy AI · Accepted Answer

To determine whether the normal approximation method can be used for testing the claim about a population proportion, we need to check the following conditions:

1. **Random Sampling**: The sample should be a random sample from the population.
2. **Sample Size**: The sample size should be large enough such that both $$ np $$ and $$ nq $$ are greater than or equal to 5, where:
   - $$ n $$ is the sample size,
   - $$ p $$ is the sample proportion (the proportion of "yes" responses),
   - $$ q = 1 - p $$ is the proportion of "no" responses.

### Given Data:
- Sample size $$ n = 521 $$
- Proportion of "yes" responses $$ p = 0.51 $$
- Proportion of "no" responses $$ q = 1 - p = 0.49 $$

### Step 1: Check the sample size conditions
We need to calculate $$ np $$ and $$ nq $$:

$$
np = n \cdot p = 521 \cdot 0.51
$$
$$
nq = n \cdot q = 521 \cdot 0.49
$$

### Step 2: Perform the calculations
Let's calculate $$ np $$ and $$ nq $$.

$$
np = 521 \cdot 0.51
$$
$$
nq = 521 \cdot 0.49
$$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$521\times 0.49$$
- step1: Multiply the numbers:
  $$255.29$$
Calculate or simplify the expression $$ 521 * 0.51 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$521\times 0.51$$
- step1: Multiply the numbers:
  $$265.71$$
### Step 2: Results of the calculations
- $$ np = 521 \cdot 0.51 \approx 265.71 $$
- $$ nq = 521 \cdot 0.49 \approx 255.29 $$

### Step 3: Check the conditions
Now we check if both $$ np $$ and $$ nq $$ are greater than or equal to 5:
- $$ np \approx 265.71 $$ (which is greater than 5)
- $$ nq \approx 255.29 $$ (which is also greater than 5)

### Step 4: Random Sampling
The problem does not provide information about whether the sample is a random sample. If the sample is not random, this condition would be violated.

### Conclusion
Based on the calculations:
- The conditions $$ np \geq 5 $$ and $$ nq \geq 5 $$ are satisfied.
- However, we cannot confirm if the sample is random based on the information provided.

### Answer to Part (d)
Given the options:
- **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 $$ np \geq 5 $$ and $$ nq \geq 5 $$ 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.

The correct answer is **D**: The sample observations are not a random sample, so a test about a population proportion using the normal approximating method cannot be used.
The sample is not a random sample, so the normal approximation method cannot be used for testing the claim.

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