To conduct a hypothesis test for the proportion, we will first calculate the test statistic and then find the p-value.
Step 1: Calculate the sample proportion \( \hat{p} \).
The sample proportion \( \hat{p} \) is calculated as: \[ \hat{p} = \frac{x}{n} \] where \( x \) is the number of successes and \( n \) is the sample size.
Given \( x = 129 \) and \( n = 744 \): \[ \hat{p} = \frac{129}{744} \approx 0.1731 \]
Step 2: Calculate the standard error (SE) of the sample proportion.
The standard error of the sample proportion is given by: \[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \] where \( p_0 \) is the proportion under the null hypothesis.
Here, \( p_0 = 0.18 \): \[ SE = \sqrt{\frac{0.18(1 - 0.18)}{744}} = \sqrt{\frac{0.18 \cdot 0.82}{744}} \approx \sqrt{\frac{0.1476}{744}} \approx \sqrt{0.0001986} \approx 0.0141 \]
Step 3: Calculate the test statistic \( z \).
The test statistic is calculated using the formula: \[ z = \frac{\hat{p} - p_0}{SE} \]
Substituting in the values we have: \[ z = \frac{0.1731 - 0.18}{0.0141} \approx \frac{-0.0069}{0.0141} \approx -0.489 \]
So the test statistic \( z \) is: \[ \text{test statistic} = -0.489 \]
Step 4: Calculate the p-value.
Since this is a left-tailed test, we will find the p-value from the standard normal distribution.
Using a standard normal distribution table or a calculator, we can find the p-value corresponding to \( z = -0.489 \).
The cumulative probability for \( z = -0.489 \) can be found as follows: \[ p\text{-value} = P(Z < -0.489) \]
Using a normal distribution calculator or table: \[ p\text{-value} \approx 0.3121 \]
Final Answers:
-
The test statistic is: \[ \text{test statistic} = -0.489 \]
-
The p-value is: \[ \text{p-value} = 0.3121 \]
1 answer