A publisher reports that 67%67% of their readers own a particular make of car.  A marketing executive wants to test the claim that the percentage is actually different from the reported percentage.  A random sample of 130130 found that 60%60% of the readers owned a particular make of car.  Is there sufficient evidence at the 0.020.02 level to support the executive's claim?

To determine if there is sufficient evidence to support the marketing executive's claim that the percentage of readers who own a particular make of car is different from the reported 67%, we need to perform a hypothesis test for proportions.

### Step 1: Set Up Hypotheses

The marketing executive claims that the percentage is different from 67%, so we will perform a **two-tailed test**.

- **Null Hypothesis (H₀)**: \( p = 0.67 \) (The population proportion is 67%)
- **Alternative Hypothesis (H₁)**: \( p \neq 0.67 \) (The population proportion is different from 67%)

### Step 2: Identify Key Information

- Reported proportion: \( p_0 = 0.67 \)
- Sample proportion: \( \hat{p} = 0.60 \)
- Sample size: \( n = 130 \)
- Significance level: \( \alpha = 0.02 \)

### Step 3: Test Statistic (Z-Score)

The test statistic for a proportion is given by the formula:

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

Now, we can calculate the test statistic:

1. \( \hat{p} = 0.60 \)
2. \( p_0 = 0.67 \)
3. \( n = 130 \)
4. \( 1 - p_0 = 0.33 \)

Now calculate the standard error:

\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.67(0.33)}{130}} \approx \sqrt{0.0017} = 0.0412
\]

Then, calculate the Z-score:

\[
z = \frac{0.60 - 0.67}{0.0412} \approx \frac{-0.07}{0.0412} \approx -1.699
\]

### Step 4: Determine the Critical Value

For a two-tailed test at the 0.02 significance level, we divide the alpha level by 2 (i.e., \( \alpha / 2 = 0.01 \)) and look up the critical z-value for 0.01 in each tail. The critical z-value for \( \alpha / 2 = 0.01 \) is approximately \( \pm 2.326 \).

### Step 5: Conclusion

- **If** \( |z| > z_{\alpha/2} \), we reject the null hypothesis.
- **If** \( |z| \leq z_{\alpha/2} \), we fail to reject the null hypothesis.

Since \( |z| = 1.699 \) is **less than** \( 2.326 \), we **fail to reject the null hypothesis**.

### Step 6: Interpretation

At the 0.02 significance level, there is **not sufficient evidence** to support the marketing executive's claim that the proportion of readers who own a particular make of car is different from the reported 67%.

---

Do you need more details or have any questions? Here are five related questions to deepen your understanding:

1. How would the results change if the sample size increased to 200?
2. What would happen if we used a one-tailed test instead?
3. How is the choice of significance level (0.02) impacting the conclusion?
4. What assumptions are we making in this hypothesis test?
5. How could this test be applied in other marketing research scenarios?

**Tip**: Always verify assumptions like random sampling and independence when conducting hypothesis tests to ensure accurate results!

A publisher reports that 67% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 130 found that 60% of the readers owned a particular make of car. Is there sufficient evidence at the 0.02 level to support the executive's claim?

This example illustrates a hypothesis test for proportions at a 0.02 significance level, testing a marketing claim that 60% of readers own a specific make of car compared to a reported 67%. Learn the step-by-step process, including calculating the Z-score and interpreting results.

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