Question 1178054: two-tailed
right-tailed
Based on a sample of 40 men, 40% owned cats
Based on a sample of 40 women, 60% owned cats The test statistic is: Correct (to 2 decimals) The critical value is: (to 2 decimals) Based on this we:
Reject the null hypothesis
Fail to reject the null hypothesis
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**1. Define the Variables**
* **p1:** Proportion of men who own cats (sample proportion p̂1 = 0.40)
* **p2:** Proportion of women who own cats (sample proportion p̂2 = 0.60)
* **n1:** Sample size of men (n1 = 40)
* **n2:** Sample size of women (n2 = 40)
**2. State the Hypotheses**
* **Null Hypothesis (H0):** p1 = p2 (There is no difference in the proportion of men and women who own cats)
* **Alternative Hypothesis (H1):** p1 ≠ p2 (There is a difference in the proportion of men and women who own cats)
Since the alternative hypothesis is "not equal to," this is a **two-tailed** test.
**3. Calculate the Test Statistic (z-score)**
* **Pooled Proportion (p̂):**
* p̂ = (x1 + x2) / (n1 + n2) = (n1 * p̂1 + n2 * p̂2) / (n1 + n2)
* p̂ = (40 * 0.40 + 40 * 0.60) / (40 + 40) = (16 + 24) / 80 = 40 / 80 = 0.50
* **Standard Error (SE):**
* SE = √[p̂(1 - p̂) * (1/n1 + 1/n2)]
* SE = √[0.50 * 0.50 * (1/40 + 1/40)] = √[0.25 * (2/40)] = √[0.25 * 0.05] = √0.0125 ≈ 0.1118
* **Test Statistic (z):**
* z = (p̂1 - p̂2) / SE
* z = (0.40 - 0.60) / 0.1118 = -0.20 / 0.1118 ≈ -1.789
Rounded to two decimal places, the test statistic is **-1.79**.
**4. Determine the Critical Value**
* **Significance Level (α):** We need a significance level to determine the critical value. Let's assume a common significance level of α = 0.05.
* **Two-Tailed Test:** For a two-tailed test with α = 0.05, the critical values are ±z(α/2) = ±z(0.025).
* **Critical Value:** From the standard normal distribution table, z(0.025) ≈ 1.96. Therefore, the critical values are ±1.96.
**5. Make a Decision**
* **Compare Test Statistic and Critical Value:**
* The test statistic (-1.79) is within the range of the critical values (-1.96 and 1.96).
* |-1.79| < 1.96
* **Decision:** Since the test statistic is not in the rejection region, we **fail to reject the null hypothesis**.
**Answers**
* **Two-tailed**
* **Test statistic:** -1.79
* **Critical value:** 1.96
* **Based on this we:** Fail to reject the null hypothesis.
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