Let's walk through the steps to solve the given problem:
### Step 1: State the Hypotheses
Null Hypothesis ([tex]$H_0$[/tex]): The population proportion [tex]\( p = 0.87 \)[/tex]
Alternative Hypothesis ([tex]$H_a$[/tex]): The population proportion [tex]\( p < 0.87 \)[/tex]
### Step 2: Determine the Significance Level
The significance level ([tex]\(\alpha\)[/tex]) is given as 0.005.
### Step 3: Sample Data
- Sample size ([tex]\(n\)[/tex]) = 289
- Number of successes ([tex]\(x\)[/tex]) = 246
### Step 4: Calculate the Sample Proportion
The sample proportion ([tex]\(\hat{p}\)[/tex]) is calculated as follows:
[tex]\[
\hat{p} = \frac{x}{n} = \frac{246}{289} \approx 0.8512
\][/tex]
### Step 5: Calculate the Test Statistic (z-score)
The test statistic can be calculated using the formula for the z-score of a sample proportion:
[tex]\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
\][/tex]
Where [tex]\( p_0 \)[/tex] is the population proportion under the null hypothesis.
Substitute the values:
[tex]\[
z = \frac{0.8512 - 0.87}{\sqrt{\frac{0.87 \times (1 - 0.87)}{289}}}
\][/tex]
After performing the calculations,
The test statistic [tex]\( z \approx -0.95 \)[/tex] (rounded to 3 decimal places).
### Step 6: Calculate the P-value
The P-value is the probability that the test statistic is less than the observed value of the test statistic under the null hypothesis. Given a z-score of [tex]\(-0.95\)[/tex], you can look up this value in a standard normal distribution table or use technology to find the P-value.
The P-value [tex]\( \approx 0.1711 \)[/tex] (rounded to 4 decimal places).
### Summary
a. Test statistic [tex]\( = -0.950 \)[/tex] (rounded to 3 decimal places)
b. P-value [tex]\( = 0.1711 \)[/tex] (rounded to 4 decimal places)
