SOLUTION: question 1: If n=330 and p' (p-prime) = 0.45, construct a 95% confidence interval. Give your answers to three decimals. question 2: Based on a sample of 35 people, the sample

Question 1177932: question 1: If n=330 and p' (p-prime) = 0.45, construct a 95% confidence interval.
Give your answers to three decimals.
question 2: Based on a sample of 35 people, the sample mean GPA was 2.97 with a standard deviation of 0.04
The p-value is:
question 3: Based on a sample of 20 men, 45% owned cats
Based on a sample of 60 women, 70% owned cats The test statistic is:
(to 2 decimals) The positive critical value is:
question 4: The sample consisted of 50 night students, with a sample mean GPA of 2.5 and a standard deviation of 0.04, and 50 day students, with a sample mean GPA of 2.52 and a standard deviation of 0.06.
the test statistic is
the p- value is

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's solve each question step-by-step.
**Question 1: Confidence Interval for a Proportion**
Given:
* n = 330
* p̂ = 0.45 (p-prime)
* Confidence level = 95%
1. **Calculate the Standard Error (SE):**
* SE = √[p̂(1 - p̂) / n]
* SE = √[0.45(0.55) / 330]
* SE = √(0.2475 / 330)
* SE = √0.00075 ≈ 0.027386
2. **Find the Critical Value (z*):**
* For a 95% confidence interval, z* ≈ 1.96
3. **Calculate the Margin of Error (ME):**
* ME = z* * SE
* ME = 1.96 * 0.027386 ≈ 0.053677
4. **Construct the Confidence Interval:**
* Confidence Interval = p̂ ± ME
* Lower Bound = 0.45 - 0.053677 ≈ 0.396323
* Upper Bound = 0.45 + 0.053677 ≈ 0.503677
5. **Round to three decimal places:**
* Lower Bound ≈ 0.396
* Upper Bound ≈ 0.504
**Answer 1:** The 95% confidence interval is (0.396, 0.504).
**Question 2: P-value for a Hypothesis Test (GPA)**
Given:
* n = 35
* Sample mean (x̄) = 2.97
* Sample standard deviation (s) = 0.04
* We are not given the null hypothesis mean. We need to know the null hypothesis mean to calculate the p-value. Assuming the Null Hypothesis is 3.0.
1. **Calculate the Standard Error (SE):**
* SE = s / √n
* SE = 0.04 / √35
* SE ≈ 0.04 / 5.9161 ≈ 0.006761
2. **Calculate the Test Statistic (t):**
* t = (x̄ - μ) / SE
* t = (2.97 - 3.0) / 0.006761
* t = -0.03 / 0.006761 ≈ -4.437
3. **Find the P-value:**
* Degrees of freedom (df) = n - 1 = 35 - 1 = 34
* Using a t-distribution table or calculator with df = 34 and t = -4.437, we find the p-value.
* For a two-tailed test, the p-value ≈ 0.00012.
**Answer 2:** The p-value is approximately 0.00012.
**Question 3: Two-Sample Proportion Test**
Given:
* Men: n1 = 20, p̂1 = 0.45
* Women: n2 = 60, p̂2 = 0.70
1. **Calculate the Pooled Proportion (p̂):**
* p̂ = (n1 * p̂1 + n2 * p̂2) / (n1 + n2)
* p̂ = (20 * 0.45 + 60 * 0.70) / (20 + 60)
* p̂ = (9 + 42) / 80 = 51 / 80 = 0.6375
2. **Calculate the Standard Error (SE):**
* SE = √[p̂(1 - p̂) * (1/n1 + 1/n2)]
* SE = √[0.6375(0.3625) * (1/20 + 1/60)]
* SE = √[0.23109375 * (0.05 + 0.016667)]
* SE = √[0.23109375 * 0.066667]
* SE = √0.01540625 ≈ 0.1241
3. **Calculate the Test Statistic (z):**
* z = (p̂1 - p̂2) / SE
* z = (0.45 - 0.70) / 0.1241
* z = -0.25 / 0.1241 ≈ -2.0145
4. **Round to two decimal places:**
* z ≈ -2.01
5. **Find the Positive Critical Value:**
* For a two-tailed test with a common significance level of α = 0.05, the critical value is z* ≈ 1.96.
**Answer 3:** The test statistic is -2.01. The positive critical value is 1.96.
**Question 4: Two-Sample T-Test (GPA)**
Given:
* Night Students: n1 = 50, x̄1 = 2.5, s1 = 0.04
* Day Students: n2 = 50, x̄2 = 2.52, s2 = 0.06
1. **Calculate the Test Statistic (t):**
* t = (x̄1 - x̄2) / √[(s1²/n1) + (s2²/n2)]
* t = (2.5 - 2.52) / √[(0.04²/50) + (0.06²/50)]
* t = -0.02 / √[(0.0016/50) + (0.0036/50)]
* t = -0.02 / √[0.000032 + 0.000072]
* t = -0.02 / √0.000104
* t = -0.02 / 0.010198
* t ≈ -1.961
2. **Find the P-value:**
* Degrees of freedom (df): Use a calculator or software to approximate the df. It would be between 50-1 and 50-1.
* Using a t-distribution calculator, with t = -1.961 and df ≈ 90, the p-value for a two-tailed test is approximately 0.053.
**Answer 4:** The test statistic is -1.961. The p-value is approximately 0.053.

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