We are tasked with testing a claim about the proportion of Americans who prefer to watch the news rather than read or listen to it. Let's follow the steps of hypothesis testing and use the P-value method.

Step 1: Define the Null and Alternative Hypotheses

Let $p$ represent the population proportion of Americans who prefer to watch the news.

• Null Hypothesis ( $H_0$ ): The null hypothesis is the claim that the population proportion is at least 50%, i.e., $p = 0.5$.

  $H_0: p \geq 0.5$

• Alternative Hypothesis ( $H_a$ ): The alternative hypothesis is the claim that fewer than half of Americans prefer to watch the news, i.e., $p < 0.5$.

  $H_a: p < 0.5$

This is a left-tailed test since we are testing if the proportion is less than 0.5.

Step 2: Calculate the Test Statistic

We will use the normal approximation to the binomial distribution since the sample size is large. The formula for the test statistic $z$ is:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Where:

• $\hat{p}$ is the sample proportion (46.9% or 0.469),
• $p_0$ is the hypothesized population proportion (0.5),
• $n$ is the sample size (1228).

Let's compute the test statistic.

The test statistic $z$ is approximately $-2.17$.

Step 3: Find the P-value

Since this is a left-tailed test, we need to find the probability that a standard normal variable is less than $-2.17$. This corresponds to the P-value, which we can determine using the standard normal distribution.

Let's compute the P-value.The P-value is approximately 0.0149.

Step 4: Decision Rule

We compare the P-value to the significance level $\alpha = 0.10$.

• If the P-value is less than $\alpha$, we reject the null hypothesis $H_0$.
• If the P-value is greater than or equal to $\alpha$, we fail to reject $H_0$.

Since the P-value (0.0149) is less than the significance level of 0.10, we reject the null hypothesis.

Conclusion:

There is sufficient evidence to support the claim that fewer than half of Americans prefer to watch the news rather than read or listen to it.

Would you like more details or have any questions?

5 Related Questions:

1. What is the significance of using the P-value method in hypothesis testing?
2. How does the normal approximation to the binomial distribution work?
3. What would the outcome be if the significance level was set to 0.05 instead?
4. Can you explain the meaning of a left-tailed test in more depth?
5. How do you interpret a P-value in hypothesis testing?

Tip: Always ensure the sample size is large enough for the normal approximation to the binomial distribution to be valid (typically $n \times p_0 \geq 5$ and $n \times (1 - p_0) \geq 5$).