Find the sample proportions and test statistic for equal proportions. Is the decision close? Find the p-value.
a. Dissatisfied workers in two companies: x1 = 40, n1 = 100, x2 = 30, n2 = 100, á = .05, two tailed test.
b. Rooms rented at least a week in advance at two hotels: x1 = 24, n1 = 200, x2 = 12, n2 = 50, á = .01, left-tailed test.
c. Home equity loan default rates in two banks: x1 = 36, n1 = 480, x2 = 26, n2 = 520, á = .05, right-tailed test.
To find the sample proportions and test statistic for equal proportions, we can follow these steps:
Step 1: Calculate the sample proportions for each group.
a. Dissatisfied workers in two companies:
- Sample proportion for Company 1: p̂1 = x1 / n1 = 40 / 100 = 0.4
- Sample proportion for Company 2: p̂2 = x2 / n2 = 30 / 100 = 0.3
b. Rooms rented at least a week in advance at two hotels:
- Sample proportion for Hotel 1: p̂1 = x1 / n1 = 24 / 200 = 0.12
- Sample proportion for Hotel 2: p̂2 = x2 / n2 = 12 / 50 = 0.24
c. Home equity loan default rates in two banks:
- Sample proportion for Bank 1: p̂1 = x1 / n1 = 36 / 480 = 0.075
- Sample proportion for Bank 2: p̂2 = x2 / n2 = 26 / 520 = 0.05
Step 2: Calculate the test statistic (Z statistic) using the formula:
Z = (p̂1 - p̂2) / √(p̂(1-p̂)(1/n1 + 1/n2))
where p̂ = (x1 + x2) / (n1 + n2)
a. Dissatisfied workers in two companies:
- p̂ = (x1 + x2) / (n1 + n2) = (40 + 30) / (100 + 100) = 70 / 200 = 0.35
Z = (0.4 - 0.3) / √(0.35 * (1-0.35) * (1/100 + 1/100))
= 0.1 / √(0.35 * 0.65 * 0.02)
≈ 0.1 / 0.073495
≈ 1.3613
b. Rooms rented at least a week in advance at two hotels:
- p̂ = (x1 + x2) / (n1 + n2) = (24 + 12) / (200 + 50) = 36 / 250 = 0.144
Z = (0.12 - 0.24) / √(0.144 * (1-0.144) * (1/200 + 1/50))
= -0.12 / √(0.144 * 0.856 * 0.0138)
≈ -0.12 / 0.048912
≈ -2.4495
c. Home equity loan default rates in two banks:
- p̂ = (x1 + x2) / (n1 + n2) = (36 + 26) / (480 + 520) = 62 / 1000 = 0.062
Z = (0.075 - 0.05) / √(0.062 * (1-0.062) * (1/480 + 1/520))
= 0.025 / √(0.062 * 0.938 * 0.001)
≈ 0.025 / 0.014259
≈ 1.752
Step 3: Determine the critical value (Z critical) based on the level of significance (á) and the type of test (two-tailed, left-tailed, or right-tailed).
a. For a two-tailed test with á = 0.05, Z critical is ±1.96 (from the standard normal distribution table).
b. For a left-tailed test with á = 0.01, Z critical is -2.33.
c. For a right-tailed test with á = 0.05, Z critical is 1.645.
Step 4: Make a decision and find the p-value.
a. Dissatisfied workers: Since the test statistic (1.3613) does not exceed the critical value (±1.96) for a two-tailed test, we fail to reject the null hypothesis. The decision is not close. To find the p-value, we can look up the Z-score of 1.3613 in the standard normal distribution table. The corresponding p-value is approximately 0.1739.
b. Rooms rented in advance: Since the test statistic (-2.4495) is less than the critical value (-2.33) for a left-tailed test, we reject the null hypothesis. The decision is close. To find the p-value, we can look up the Z-score of -2.4495 in the standard normal distribution table. The corresponding p-value is approximately 0.0070.
c. Home equity loan default rates: Since the test statistic (1.752) exceeds the critical value (1.645) for a right-tailed test, we reject the null hypothesis. The decision is close. To find the p-value, we can look up the Z-score of 1.752 in the standard normal distribution table. The corresponding p-value is approximately 0.0398.