Sure, let's go step by step to calculate the test statistic for the given hypotheses:
### Step 1: Formulate the hypotheses
The null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_a\)[/tex]) are:
[tex]\[ H_0: p = 0.09 \][/tex]
[tex]\[ H_a: p > 0.09 \][/tex]
### Step 2: Collect data
We have:
- Sample size ([tex]\(n\)[/tex]): 87
- Number of returned items: 7
### Step 3: Calculate the sample proportion ([tex]\(\hat{p}\)[/tex])
The sample proportion ([tex]\(\hat{p}\)[/tex]) is the number of returned items divided by the sample size:
[tex]\[ \hat{p} = \frac{7}{87} \][/tex]
Substituting the values:
[tex]\[ \hat{p} = 0.08045977011494253 \][/tex]
### Step 4: Calculate the standard error (SE) of the sampling distribution of the proportion
The standard error can be calculated using the formula:
[tex]\[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
where:
- [tex]\(p_0\)[/tex] is the population proportion (0.09)
- [tex]\(n\)[/tex] is the sample size (87)
Substituting the values:
[tex]\[ SE = \sqrt{\frac{0.09 \times (1 - 0.09)}{87}} = \sqrt{\frac{0.09 \times 0.91}{87}} \][/tex]
Calculating this gives:
[tex]\[ SE = 0.0306819052593679 \][/tex]
### Step 5: Calculate the test statistic (z-value)
The z-value is calculated using the formula:
[tex]\[ z = \frac{\hat{p} - p_0}{SE} \][/tex]
Substituting the known values:
[tex]\[ z = \frac{0.08045977011494253 - 0.09}{0.0306819052593679} \][/tex]
This gives:
[tex]\[ z = -0.31093994340995545 \][/tex]
### Step 6: Round the test statistic to 3 decimal places
Finally, rounding the z-value to 3 decimal places:
[tex]\[ z \approx -0.311 \][/tex]
### Conclusion
The calculated test statistic is [tex]\(-0.311\)[/tex] when rounded to three decimal places. This completes the step-by-step solution for calculating the test statistic by hand.
