Question

Two community colleges want to compare student satisfication at their respective schools. Rose Petal Community College does a random sample of 74 students and find that 58 say they are "satisfied" or "extremely satisfied" with the school. Archimedes Spiral Community College does a random sample of 108 students and finds that 91 say they are "satisfied" or better with the school.

Using Rose Petal CC as , and Archimedes Spiral CC as P2, construct a 95% confidence interval for the difference in proportions. Use three decimals when computing the sample proportions, and give your final answer to three decimal places.
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A random sample of 230 people living in Seattle found that 83 of them had a pet. A random sample of 172 people in Spokane found that 47 of them had a pet. Let P, be the proportion of pet-owners in Seattle and , be the proportion of pet owners in Spokane. Found an 80% confidence interval for the difference in proportions.
Use three decimals when computing the proportions and give your final answer to three decimal places.
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Based on this confidence interval, what could we conclude from the samples:
A. At 80% confidence, the proportion of pet owners in Seatle is smaller than the proportion of pet owners in Spokane
B. At 80% confidence, the proportion of pet owners in Seattle is greater than the proportion of pet owners in Spokane
C. At 80% confidence, the proportion of pet-owners in Seattle and Spokane is the same

Answer 1

To construct a 95% confidence interval for the difference in proportions of student satisfaction between Rose Petal CC and Archimedes Spiral CC, we can use the formula (P1 - P2) ± Z * √[(P1 * (1 - P1) / n1) + (P2 * (1 - P2) / n2)]. Based on the given sample proportions, sample sizes, and Z-value for 95% confidence, the 95% confidence interval for the difference in proportions is approximately -0.076 to -0.042. This means that the proportion of students satisfied at Archimedes Spiral CC is significantly higher than at Rose Petal CC at 95% confidence.

Step-by-step explanation:

To construct a 95% confidence interval for the difference in proportions between Rose Petal Community College (P1) and Archimedes Spiral Community College (P2), we can use the formula:

(P1 - P2) ± Z * √[(P1 * (1 - P1) / n1) + (P2 * (1 - P2) / n2)],

where:

• P1 = proportion of students satisfied at Rose Petal CC
• P2 = proportion of students satisfied at Archimedes Spiral CC
• n1 = sample size at Rose Petal CC
• n2 = sample size at Archimedes Spiral CC
• Z = Z-value for the desired confidence level (0.95)

Using the given information, we have:

• P1 = 58/74 = 0.784
• P2 = 91/108 = 0.843
• n1 = 74
• n2 = 108
• Z = 1.96 (for 95% confidence)

Plugging in these values into the formula, we get:

(0.784 - 0.843) ± 1.96 * √[(0.784 * (1 - 0.784) / 74) + (0.843 * (1 - 0.843) / 108)]

This simplifies to:

-0.059 ± 1.96 * √[0.003 / 74 + 0.004 / 108]

Calculating further, we get:

-0.059 ± 1.96 * √(0.000040 + 0.000037)

-0.059 ± 1.96 * √0.000077

-0.059 ± 1.96 * 0.00877

-0.059 ± 0.01714

Therefore, the 95% confidence interval for the difference in proportions is approximately -0.076 to -0.042.

Based on this interval, we can conclude that at 95% confidence, the proportion of students satisfied at Archimedes Spiral CC is significantly higher than at Rose Petal CC.