Question

select the correct answer.
a restaurant owner wants to determine the effectiveness of his servers. the owner conducted a survey by asking 30 customers about the servers’ effectiveness, on a day when the restaurant had 450 customers. the owner found that 65% of those surveyed were satisfied with the service offered.
assuming a 95% confidence level, which statement holds true?
a. as the sample size is appropriately large, the margin of error is 0.203.

b. as the sample size is appropriately large, the margin of error is 0.171.

c. as the sample size is too small, the margin of error cannot be trusted.

d. as the sample size is too small, the margin of error is 0.203.

Explanation:

Step1: Check Sample Size Appropriateness

The total number of customers is 450, and the sample size is 30. A sample size of 30 is generally considered the threshold for the Central Limit Theorem to apply (for approximate normality in sampling distributions). However, when the population is 450, we can also check if \( n \leq 0.05N \) (5% of population) for independence. \( 0.05\times450 = 22.5 \), and \( 30>22.5 \), so the sample size might be considered too large relative to the population for the "10% rule" (independence assumption), but more importantly, for margin of error calculation, the formula for margin of error (ME) for a proportion is \( ME = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \). For 95% confidence, \( z_{\alpha/2}=1.96 \), \( \hat{p}=0.65 \), \( n = 30 \).

Step2: Calculate Margin of Error

First, compute \( \hat{p}(1 - \hat{p})=0.65\times0.35 = 0.2275 \). Then, \( \frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.2275}{30}\approx0.007583 \). Take the square root: \( \sqrt{0.007583}\approx0.0871 \). Multiply by \( z_{\alpha/2}=1.96 \): \( 1.96\times0.0871\approx0.1707\approx0.171 \). But wait, the sample size of 30: is it "appropriately large"? The Central Limit Theorem (CLT) for proportions is often considered to apply when \( np\geq5 \) and \( n(1 - p)\geq5 \). Here, \( np = 30\times0.65 = 19.5\geq5 \), \( n(1 - p)=30\times0.35 = 10.5\geq5 \), so CLT applies, sample size is appropriately large? Wait, no—earlier, the 10% rule: \( n = 30 \), \( N = 450 \), \( 30/450\approx0.0667>0.05 \), so the independence assumption (that sampling without replacement doesn't affect the proportion much) is violated. But the options talk about "sample size too small" or "appropriately large". Wait, the options C and D say "sample size too small". Wait, maybe the key is that when \( n = 30 \), for the CLT, it's a common threshold, but if the population is 450, and sample is 30 (more than 5% of population), the sample size is too large for the 10% rule, but the margin of error calculation: but let's re - evaluate the options. Wait, the options C and D say "sample size too small". Wait, maybe the problem is that 30 is considered small? No, CLT for proportions needs \( np\geq5 \) and \( n(1 - p)\geq5 \), which is satisfied here. But the options: A says "appropriately large, ME = 0.203", B says "appropriately large, ME = 0.171", C says "too small, ME cannot be trusted", D says "too small, ME = 0.203". Wait, maybe the mistake is in the 10% rule. If we ignore the 10% rule (some sources say it's a guideline, not strict), then CLT applies. But let's recalculate ME with the finite population correction (FPC) factor, since \( n/N>0.05 \). The FPC factor is \( \sqrt{\frac{N - n}{N - 1}}=\sqrt{\frac{450 - 30}{450 - 1}}=\sqrt{\frac{420}{449}}\approx\sqrt{0.9354}\approx0.9672 \). Then, the standard error with FPC is \( 0.0871\times0.9672\approx0.0843 \), and ME is \( 1.96\times0.0843\approx0.1652\approx0.165 \), but the options don't have that. Wait, maybe the question is considering that 30 is "too small" because when the population is 450, sampling 30 (about 6.67% of population) is too large for the 10% rule, so the margin of error calculation without FPC is inaccurate, and thus the margin of error cannot be trusted? Wait, no—let's check the options again. Option B: "As the sample size is appropriately large, the margin of error is 0.171". We calculated ME ≈ 0.171 with CLT (no FPC). But the 10% rule: if we follow the 10% rule, then since \( n/N>0.05 \), we should use FPC, but maybe the question ignores FPC. Also, "appropriately large"—CLT applies (np and n(1 - p) ≥5), so sample size is appropriately large. So then ME is \( 1.96\sqrt{\frac{0.65\times0.35}{30}}\approx1.96\sqrt{0.007583}\approx1.96\times0.0871\approx0.171 \), which matches option B. Wait, but earlier I thought about FPC, but maybe the question considers the sample size "appropriately large" because CLT conditions are met (np and n(1 - p) ≥5), so option B. But wait, let's check the other options. Option C: "As the sample size is too small, the margin of error cannot be trusted". But np = 19.5, n(1 - p)=10.5, both ≥5, so CLT applies, sample size is not too small. So option B is correct? Wait, no—wait the total customers are 450, sample is 30. The 10% rule: sample size should be ≤10% of population for independence. 10% of 450 is 45, and 30 ≤45, so 30 is ≤10% of 450, so the 10% rule is satisfied (I made a mistake earlier: 5% rule is for some, 10% is more common). So 30 ≤0.1×450 = 45, so independence assumption holds. Then, CLT applies (np and n(1 - p) ≥5), so sample size is appropriately large. Then ME is ≈0.171, so option B. Wait, but let's recalculate:

\( \hat{p}=0.65 \), \( n = 30 \), \( z = 1.96 \) (95% confidence).

Standard error \( SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.65\times0.35}{30}}=\sqrt{\frac{0.2275}{30}}\approx\sqrt{0.007583}\approx0.0871 \)

Margin of error \( ME = z\times SE=1.96\times0.0871\approx0.1707\approx0.171 \)

So since the sample size is appropriately large (CLT conditions met, and 10% rule met as 30 ≤45), the margin of error is 0.171, so option B.

Wait, but the options A and B say "appropriately large", A has ME 0.203, B has 0.171. Our calculation is 0.171, so B is correct.

Answer:

B. As the sample size is appropriately large, the margin of error is 0.171.