Sure, let's go through the steps to calculate the test statistics for this hypothesis test.
### Step 1: Gather the given information
- Sample size, [tex]\( n \)[/tex]: 136
- Number of successes, [tex]\( x \)[/tex]: 76
- Null hypothesis proportion, [tex]\( p_0 \)[/tex]: 0.53
### Step 2: Calculate the sample proportion
The sample proportion, [tex]\( \hat{p} \)[/tex], is calculated as:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{76}{136} = 0.5588 \][/tex]
### Step 3: Calculate the standard error
The standard error (SE) of the sample proportion under the null hypothesis is:
[tex]\[ \text{SE} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.53 \times (1 - 0.53)}{136}} = 0.0428 \][/tex]
### Step 4: Calculate the test statistic (z-score)
The test statistic (z) is calculated using the formula:
[tex]\[ z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.5588 - 0.53}{0.0428} = 0.673 \][/tex]
### Final Answer
So, the test statistic rounded to 3 decimal places is:
[tex]\[ z = 0.673 \][/tex]
We have systematically calculated the test statistic for the given hypothesis test.
