Sure, let's solve this step by step.
### Step 1: Formulate the Hypotheses
We are given the null hypothesis and the alternative hypothesis as follows:
- [tex]\( H_0: p = 0.89 \)[/tex]
- [tex]\( H_1: p > 0.89 \)[/tex]
### Step 2: Gather the Data
We have the following sample information:
- Sample size, [tex]\( n = 708 \)[/tex]
- Number of successes, [tex]\( x = 648 \)[/tex]
### Step 3: Calculate the Sample Proportion
The sample proportion, [tex]\( \hat{p} \)[/tex], is computed as:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{648}{708} \approx 0.915 \][/tex]
### Step 4: Calculate the Standard Error
The standard error (SE) under the null hypothesis is given by:
[tex]\[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
where [tex]\( p_0 \)[/tex] is the hypothesized population proportion (in this case, 0.89). Substituting the values:
[tex]\[ SE = \sqrt{\frac{0.89 \times (1 - 0.89)}{708}} = \sqrt{\frac{0.89 \times 0.11}{708}} \approx 0.012 \][/tex]
### Step 5: Calculate the Test Statistic (Z-score)
The test statistic for a one-sample proportion test is calculated as:
[tex]\[ Z = \frac{\hat{p} - p_0}{SE} \][/tex]
Substituting in the values:
[tex]\[ Z = \frac{0.915 - 0.89}{0.012} \approx 2.148 \][/tex]
So, the test statistic is:
[tex]\[ \text{test statistic} = 2.148 \][/tex]
### Step 6: Calculate the P-value
The p-value corresponds to the area to the right of the calculated Z-score in the standard normal distribution. Using the standard normal table or a statistical calculator:
[tex]\[ \text{p-value} = 1 - \Phi(Z) \][/tex]
where [tex]\( \Phi(Z) \)[/tex] is the cumulative distribution function of the standard normal distribution at [tex]\( Z = 2.148 \)[/tex].
[tex]\[ \text{p-value} \approx 1 - 0.9841 = 0.0159 \][/tex]
So, the p-value is:
[tex]\[ \text{p-value} = 0.0159 \][/tex]
### Step 7: Compare the P-value to the Significance Level
We compare the p-value to the significance level [tex]\( \alpha = 0.005 \)[/tex].
Since [tex]\( 0.0159 > 0.005 \)[/tex], the p-value is greater than [tex]\( \alpha \)[/tex].
### Conclusion
The test statistic is 2.148, the p-value is 0.0159, and since the p-value is greater than the significance level [tex]\( \alpha = 0.005 \)[/tex], we do not reject the null hypothesis.
In summary:
- Test statistic [tex]\( = 2.148 \)[/tex]
- P-value [tex]\( = 0.0159 \)[/tex]
- The p-value is greater than [tex]\( \alpha \)[/tex]
