Final answer:
The test statistic (-1.85) is less than the critical value (-1.645), we can reject the null hypothesis.
And, there is sufficient evidence to support the claim that the proportion of Americans who approve the president's management of the crisis is less than 50%.
Step-by-step explanation:
To test the claim that the proportion of Americans who approve the president's management of the crisis is less than 50%, we can use a hypothesis test.
Step 1: Set up the null hypothesis and alternative hypothesis.
- Null hypothesis (H0): The proportion of Americans who approve the president's management of the crisis is equal to 50%.
- Alternative hypothesis (Ha): The proportion of Americans who approve the president's management of the crisis is less than 50%.
Step 2: Determine the significance level (α).
The significance level is given as 5%, which corresponds to α = 0.05.
Step 3: Calculate the test statistic and the critical value.
In this case, the test statistic is the z-statistic, and we can calculate it using the formula:
z = (p - P) / sqrt(P(1-P)/n)
where:
- p is the sample proportion (246/506 = 0.486)
- P is the hypothesized proportion (0.50)
- n is the sample size (506)
Calculating the z-statistic, we get:
z = (0.486 - 0.50) / sqrt(0.50(1-0.50)/506) = -1.85
Using a z-table or a calculator, we can find that the critical value for a one-tailed test at α = 0.05 is approximately -1.645.
Step 4: Make a decision based on the test statistic and critical value.
Since the test statistic (-1.85) is less than the critical value (-1.645), we can reject the null hypothesis.
Step 5: Interpret the results.
There is sufficient evidence to support the claim that the proportion of Americans who approve the president's management of the crisis is less than 50%.