To solve this problem, let's go through the steps methodically:
### Step 1: Understand the Hypotheses and Test Statistic
We are given the alternative hypothesis [tex]\( H_1: p < \frac{3}{5} \)[/tex], and the test statistic [tex]\( z = -1.68 \)[/tex].
### Step 2: Determine the P-Value
The P-value is the probability that the test statistic would take a value as extreme as, or more extreme than, the one actually observed, assuming that the null hypothesis [tex]\( H_0 \)[/tex] is true.
Given our test statistic [tex]\( z = -1.68 \)[/tex], we need to find the corresponding P-value. The P-value for a z-score in a standard normal distribution can be found using the cumulative distribution function (CDF). For [tex]\( z = -1.68 \)[/tex], the P-value is the area to the left of [tex]\( z = -1.68 \)[/tex] on the standard normal distribution.
### Step 3: P-Value Calculation
The P-value for [tex]\( z = -1.68 \)[/tex] is approximately [tex]\( 0.0465 \)[/tex].
### Step 4: Compare the P-Value with the Significance Level
Here, we compare the computed P-value with the given significance level [tex]\( \alpha = 0.05 \)[/tex]:
- If the P-value is less than the significance level ([tex]\( P < 0.05 \)[/tex]), we reject the null hypothesis.
- If the P-value is greater than or equal to the significance level ([tex]\( P \geq 0.05 \)[/tex]), we fail to reject the null hypothesis.
Given the P-value [tex]\( \approx 0.0465 \)[/tex] and the significance level [tex]\( \alpha = 0.05 \)[/tex]:
- Because [tex]\( 0.0465 < 0.05 \)[/tex], we reject the null hypothesis.
### Step 5: Conclusion
Since the P-value [tex]\( 0.0465 \)[/tex] is less than the significance level [tex]\( 0.05 \)[/tex], we reject the null hypothesis.
### Final Answer
- [tex]\( 0.0465 \)[/tex]; reject the null hypothesis
Therefore, the correct choice from the given options is:
[tex]\[ 0.0465 \; \text{; reject the null hypothesis} \][/tex]
