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Answer :
We want to test the claim that more than [tex]$25\%$[/tex] of adults describe themselves as organized. We set up our hypotheses as
[tex]$$
H_0: p = 0.25 \quad \text{and} \quad H_a: p > 0.25.
$$[/tex]
Below is a step-by-step solution.
1. Conditions for Inference
- Random: The sample of [tex]$100$[/tex] adults is a random sample.
- 10% Condition: The sample size of [tex]$100$[/tex] is less than [tex]$10\%$[/tex] of the adult population, so this condition is met.
- Large Counts Condition: We calculate the expected counts under the null hypothesis:
[tex]$$
np_0 = 100 \times 0.25 = 25,
$$[/tex]
[tex]$$
n(1-p_0)= 100 \times 0.75 = 75.
$$[/tex]
Both [tex]$25$[/tex] and [tex]$75$[/tex] are greater than or equal to [tex]$10$[/tex], so this condition is satisfied.
2. Sample Proportion
In the sample, [tex]$42$[/tex] out of [tex]$100$[/tex] adults describe themselves as organized. The sample proportion is
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
3. Standard Error of the Sampling Distribution
Under the null hypothesis, the standard error is
[tex]$$
\text{SE} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{100}} \approx 0.0433.
$$[/tex]
4. Test Statistic
The [tex]$z$[/tex]-score is computed by
[tex]$$
z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.42 - 0.25}{0.0433} \approx 3.93.
$$[/tex]
5. P-value
Since this is a right-tailed test, the p-value is
[tex]$$
p\text{-value} = 1 - \Phi(z),
$$[/tex]
where [tex]$\Phi(z)$[/tex] is the cumulative distribution function of the standard normal distribution. With [tex]$z \approx 3.93$[/tex], the p-value is approximately
[tex]$$
4.32 \times 10^{-5}.
$$[/tex]
6. Conclusion
With a significance level of [tex]$\alpha = 0.01$[/tex], we compare the p-value to [tex]$\alpha$[/tex]. Since
[tex]$$
4.32 \times 10^{-5} < 0.01,
$$[/tex]
we reject [tex]$H_0$[/tex]. This provides convincing evidence that more than [tex]$25\%$[/tex] of adults describe themselves as organized.
7. Summary of Checks for Inference
- Random: We have a random sample.
- 10% Condition: [tex]$100$[/tex] adults is less than [tex]$10\%$[/tex] of the population.
- Large Counts:
[tex]$$np_0 = 25,$$[/tex]
[tex]$$n(1-p_0) = 75,$$[/tex]
both are at least [tex]$10$[/tex].
Thus, the conditions for inference are met, and the data provide convincing evidence at the [tex]$\alpha = 0.01$[/tex] level that greater than [tex]$25\%$[/tex] of adults describe themselves as organized.
[tex]$$
H_0: p = 0.25 \quad \text{and} \quad H_a: p > 0.25.
$$[/tex]
Below is a step-by-step solution.
1. Conditions for Inference
- Random: The sample of [tex]$100$[/tex] adults is a random sample.
- 10% Condition: The sample size of [tex]$100$[/tex] is less than [tex]$10\%$[/tex] of the adult population, so this condition is met.
- Large Counts Condition: We calculate the expected counts under the null hypothesis:
[tex]$$
np_0 = 100 \times 0.25 = 25,
$$[/tex]
[tex]$$
n(1-p_0)= 100 \times 0.75 = 75.
$$[/tex]
Both [tex]$25$[/tex] and [tex]$75$[/tex] are greater than or equal to [tex]$10$[/tex], so this condition is satisfied.
2. Sample Proportion
In the sample, [tex]$42$[/tex] out of [tex]$100$[/tex] adults describe themselves as organized. The sample proportion is
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
3. Standard Error of the Sampling Distribution
Under the null hypothesis, the standard error is
[tex]$$
\text{SE} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{100}} \approx 0.0433.
$$[/tex]
4. Test Statistic
The [tex]$z$[/tex]-score is computed by
[tex]$$
z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.42 - 0.25}{0.0433} \approx 3.93.
$$[/tex]
5. P-value
Since this is a right-tailed test, the p-value is
[tex]$$
p\text{-value} = 1 - \Phi(z),
$$[/tex]
where [tex]$\Phi(z)$[/tex] is the cumulative distribution function of the standard normal distribution. With [tex]$z \approx 3.93$[/tex], the p-value is approximately
[tex]$$
4.32 \times 10^{-5}.
$$[/tex]
6. Conclusion
With a significance level of [tex]$\alpha = 0.01$[/tex], we compare the p-value to [tex]$\alpha$[/tex]. Since
[tex]$$
4.32 \times 10^{-5} < 0.01,
$$[/tex]
we reject [tex]$H_0$[/tex]. This provides convincing evidence that more than [tex]$25\%$[/tex] of adults describe themselves as organized.
7. Summary of Checks for Inference
- Random: We have a random sample.
- 10% Condition: [tex]$100$[/tex] adults is less than [tex]$10\%$[/tex] of the population.
- Large Counts:
[tex]$$np_0 = 25,$$[/tex]
[tex]$$n(1-p_0) = 75,$$[/tex]
both are at least [tex]$10$[/tex].
Thus, the conditions for inference are met, and the data provide convincing evidence at the [tex]$\alpha = 0.01$[/tex] level that greater than [tex]$25\%$[/tex] of adults describe themselves as organized.
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