https://file.aiquickdraw.com/mathbot/file/1730385353864fcj2adhw.jpg

To solve this hypothesis testing question, let’s go through the details and steps involved.

### Given Information:
- **Claimed proportion** (\( p_0 \)): 0.25 (25%)
- **Sample size** (\( n \)): 123
- **Observed number of successes** (\( x \)): 30 (people who would watch one of the TV shows)
- **Alternative Hypothesis**: \( H_a : p < 0.25 \)

### Step 1: Set up hypotheses
The hypotheses are:
- **Null Hypothesis** (\( H_0 \)): \( p = 0.25 \)
- **Alternative Hypothesis** (\( H_a \)): \( p < 0.25 \)

### Step 2: Calculate sample proportion
The sample proportion (\( \hat{p} \)) is calculated as follows:
\[
\hat{p} = \frac{x}{n} = \frac{30}{123} \approx 0.244
\]

### Step 3: Calculate the standard error
The standard error (\( SE \)) for a proportion is given by:
\[
SE = \sqrt{\frac{p_0 (1 - p_0)}{n}}
\]
Substitute \( p_0 = 0.25 \) and \( n = 123 \):
\[
SE = \sqrt{\frac{0.25 \times (1 - 0.25)}{123}} = \sqrt{\frac{0.25 \times 0.75}{123}} \approx 0.0387
\]

### Step 4: Calculate the test statistic (\( z \)-score)
The \( z \)-score for testing the proportion is calculated as:
\[
z = \frac{\hat{p} - p_0}{SE}
\]
Substitute \( \hat{p} = 0.244 \), \( p_0 = 0.25 \), and \( SE = 0.0387 \):
\[
z = \frac{0.244 - 0.25}{0.0387} \approx -0.155
\]

### Final Answer:
The test statistic, rounded to three decimal places, is:
\[
z \approx -0.155
\]

Would you like additional details or have any questions about this process?

---

### Related Questions:
1. How is the critical value for a left-tailed test determined in hypothesis testing?
2. What are the implications of a negative \( z \)-score in hypothesis testing?
3. How does sample size affect the standard error in hypothesis tests?
4. What would the hypothesis test conclusion be if the calculated \( z \)-value was less than the critical value?
5. How is a p-value calculated from a \( z \)-score, and what does it represent?

### Tip:
In hypothesis testing, always check if your test is one-tailed or two-tailed, as it affects the critical values and interpretation of results.

In a recent Super Bowl, a TV network predicted that 25% of the audience would express an interest in seeing one of its forthcoming television shows. The network ran commercials for these shows during the Super Bowl. The day after the Super Bowl, an Advertising Group sampled 123 people who saw the commercials and found that 30 of them said they would watch one of the television shows. Suppose you are testing the following null and alternative hypotheses:

H₀: p = 0.25
Hₐ: p < 0.25

Calculate the test statistic, rounded to 3 decimal places.

Solve a hypothesis test for a proportion using data from a Super Bowl ad campaign. Given a sample of 123 people with 30 expressing interest, calculate the test statistic for testing if the true proportion is less than 25%. This problem involves concepts of hypothesis testing, proportion testing, standard error, and Z-score calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation