To determine if the marketing manager's claim is valid, we need to conduct a hypothesis test for a population proportion.

Step 1: State the Hypotheses

The null and alternate hypotheses are structured as follows:

• $H_0$ (Null hypothesis): The proportion of children aged 8–12 with cell phones is greater than or equal to 59%.
  • $H_0: p \geq 0.59$
• $H_1$ (Alternate hypothesis): The proportion of children aged 8–12 with cell phones is less than 59%.
  • $H_1: p < 0.59$

This is a left-tailed test because the claim suggests that the proportion is less than 59%.

Step 2: Significance Level

The significance level is given as $\alpha = 0.01$.

Step 3: Sample Proportion

The sample size $n = 822$, and the number of children with cell phones $x = 460$.

The sample proportion $\hat{p}$ is calculated as: $\hat{p} = \frac{x}{n} = \frac{460}{822} \approx 0.5595$

Step 4: Test Statistic

The test statistic for a population proportion is calculated as: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$ Where:

• $\hat{p}$ is the sample proportion
• $p_0 = 0.59$ is the hypothesized population proportion
• $n = 822$ is the sample size

Substituting in the values: $z = \frac{0.5595 - 0.59}{\sqrt{\frac{0.59(1 - 0.59)}{822}}}$ $z = \frac{-0.0305}{\sqrt{\frac{0.59 \times 0.41}{822}}} = \frac{-0.0305}{\sqrt{0.000294}} \approx \frac{-0.0305}{0.01714} \approx -1.78$

Step 5: Critical Value

Since this is a left-tailed test and $\alpha = 0.01$, we find the critical value from the standard normal distribution table (z-table). The critical value for $\alpha = 0.01$ in a left-tailed test is -2.33.

Step 6: Conclusion

• The test statistic $z \approx -1.78$ is not less than the critical value of $z = -2.33$.
• Therefore, we fail to reject the null hypothesis $H_0$.

Final Conclusion:

There is not enough evidence at the $\alpha = 0.01$ level of significance to support the marketing manager's claim that less than 59% of children aged 8–12 have cell phones.

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Would you like more details on any step or have questions?

Here are 5 related questions to consider:

1. What is the significance of using a left-tailed test in this scenario?
2. How does the choice of $\alpha = 0.01$ affect the results?
3. How would the conclusion change if the test statistic was smaller than the critical value?
4. What happens if we choose a different level of significance, such as $\alpha = 0.05$?
5. Can we calculate a confidence interval for the sample proportion?

Tip: When conducting hypothesis tests, always double-check the direction of the test (left-tailed, right-tailed, or two-tailed) to ensure you're using the correct critical values.