To calculate the test statistic and p-value for the given problem, we will follow the steps outlined below.
Step 1: Calculate the Sample Proportion
The sample proportion (\( \hat{p} \)) is calculated as follows:
\[ \hat{p} = \frac{x}{n} \]
Where:
- \( x = 62 \) (number of successes)
- \( n = 167 \) (sample size)
So,
\[ \hat{p} = \frac{62}{167} \approx 0.3713 \]
Step 2: Calculate the Standard Error (SE)
The standard error of the sample proportion can be calculated using the formula:
\[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \]
Where \( p_0 = 0.33 \) is the population proportion under the null hypothesis.
Calculating \( SE \):
\[ SE = \sqrt{\frac{0.33 \times (1 - 0.33)}{167}} = \sqrt{\frac{0.33 \times 0.67}{167}} = \sqrt{\frac{0.2211}{167}} \approx \sqrt{0.001324} \approx 0.0364 \]
Step 3: Calculate the Test Statistic
The test statistic (\( z \)) is calculated using the formula:
\[ z = \frac{\hat{p} - p_0}{SE} \]
Substituting the values we have:
\[ z = \frac{0.3713 - 0.33}{0.0364} \approx \frac{0.0413}{0.0364} \approx 1.136 \]
So the test statistic is approximately:
\[ \text{test statistic} = 1.136 \]
Step 4: Calculate the p-value
Since we are conducting a one-tailed test (right-tailed), we will find the p-value corresponding to the calculated \( z \) statistic from the standard normal distribution. The p-value can be found using a z-table or a calculator.
Using standard normal distribution tables (or a calculator), we find:
- Find the area to the left of \( z = 1.136 \).
- Subtract this value from 1 to get the area to the right (p-value).
Using a z-table or calculator, we find:
\[ P(Z < 1.136) \approx 0.8714 \]
Therefore,
\[ \text{p-value} = 1 - 0.8714 = 0.1286 \]
Final Answers
Thus, the final results are:
- Test statistic: 1.136
- p-value: 0.1286
Test statistic = 1.136
p-value = 0.1286
1 answer