Answer :
We are testing the hypotheses
[tex]$$
H_0: p = 0.25 \quad \text{versus} \quad H_a: p > 0.25,
$$[/tex]
where [tex]$p$[/tex] is the true proportion of adults who describe themselves as organized. A random sample of 100 adults yielded 42 adults who are organized, so the sample proportion is
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
Below is a step‐by‐step explanation.
──────────────────────────────
Step 1. Check the Conditions for Inference
1. Randomness Condition:
The data come from a random sample of 100 adults.
2. 10% Condition:
The sample size of 100 is less than 10% of the entire population of adults (which is usually very large), ensuring independence.
3. Large Counts Condition:
We need to have both
[tex]$$
n p_0 \quad \text{and} \quad n(1-p_0)
$$[/tex]
be at least 10. Here,
[tex]$$
n p_0 = 100 \times 0.25 = 25,
$$[/tex]
[tex]$$
n(1-p_0) = 100 \times 0.75 = 75.
$$[/tex]
Both 25 and 75 are clearly at least 10.
Thus, all conditions for inference are met.
──────────────────────────────
Step 2. Compute the Test Statistic
The standard error under the null hypothesis is given by
[tex]$$
SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25\times0.75}{100}} \approx 0.0433.
$$[/tex]
The test statistic ([tex]$z$[/tex]-score) is
[tex]$$
z = \frac{\hat{p} - p_0}{SE} = \frac{0.42 - 0.25}{0.0433} \approx 3.93.
$$[/tex]
──────────────────────────────
Step 3. Find the p-value
Since the alternative hypothesis is [tex]$p > 0.25$[/tex], we are dealing with a right-tailed test. The [tex]$p$[/tex]-value is the probability of obtaining a [tex]$z$[/tex]-score of 3.93 or above. This value is extremely small:
[tex]$$
\text{p-value} \approx 0.0000432.
$$[/tex]
──────────────────────────────
Step 4. Make a Decision
At a significance level of [tex]$\alpha=0.01$[/tex], we compare the p-value to [tex]$\alpha$[/tex]. Since
[tex]$$
0.0000432 < 0.01,
$$[/tex]
we reject the null hypothesis. This provides convincing evidence that the proportion of adults who describe themselves as organized is greater than [tex]$25\%$[/tex].
──────────────────────────────
Summary of the Conditions:
- Random: We have a random sample of 100 adults.
- 10% Condition: 100 adults is less than 10% of the entire adult population.
- Large Counts:
- [tex]$n p_0 = 25$[/tex],
- [tex]$n(1-p_0) = 75$[/tex],
and both values are at least 10.
Thus, all the conditions for inference are satisfied.
──────────────────────────────
Final Conclusion:
The data provide convincing evidence at the [tex]$\alpha=0.01$[/tex] level that greater than [tex]$25\%$[/tex] of adults would describe themselves as organized.
[tex]$$
H_0: p = 0.25 \quad \text{versus} \quad H_a: p > 0.25,
$$[/tex]
where [tex]$p$[/tex] is the true proportion of adults who describe themselves as organized. A random sample of 100 adults yielded 42 adults who are organized, so the sample proportion is
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
Below is a step‐by‐step explanation.
──────────────────────────────
Step 1. Check the Conditions for Inference
1. Randomness Condition:
The data come from a random sample of 100 adults.
2. 10% Condition:
The sample size of 100 is less than 10% of the entire population of adults (which is usually very large), ensuring independence.
3. Large Counts Condition:
We need to have both
[tex]$$
n p_0 \quad \text{and} \quad n(1-p_0)
$$[/tex]
be at least 10. Here,
[tex]$$
n p_0 = 100 \times 0.25 = 25,
$$[/tex]
[tex]$$
n(1-p_0) = 100 \times 0.75 = 75.
$$[/tex]
Both 25 and 75 are clearly at least 10.
Thus, all conditions for inference are met.
──────────────────────────────
Step 2. Compute the Test Statistic
The standard error under the null hypothesis is given by
[tex]$$
SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25\times0.75}{100}} \approx 0.0433.
$$[/tex]
The test statistic ([tex]$z$[/tex]-score) is
[tex]$$
z = \frac{\hat{p} - p_0}{SE} = \frac{0.42 - 0.25}{0.0433} \approx 3.93.
$$[/tex]
──────────────────────────────
Step 3. Find the p-value
Since the alternative hypothesis is [tex]$p > 0.25$[/tex], we are dealing with a right-tailed test. The [tex]$p$[/tex]-value is the probability of obtaining a [tex]$z$[/tex]-score of 3.93 or above. This value is extremely small:
[tex]$$
\text{p-value} \approx 0.0000432.
$$[/tex]
──────────────────────────────
Step 4. Make a Decision
At a significance level of [tex]$\alpha=0.01$[/tex], we compare the p-value to [tex]$\alpha$[/tex]. Since
[tex]$$
0.0000432 < 0.01,
$$[/tex]
we reject the null hypothesis. This provides convincing evidence that the proportion of adults who describe themselves as organized is greater than [tex]$25\%$[/tex].
──────────────────────────────
Summary of the Conditions:
- Random: We have a random sample of 100 adults.
- 10% Condition: 100 adults is less than 10% of the entire adult population.
- Large Counts:
- [tex]$n p_0 = 25$[/tex],
- [tex]$n(1-p_0) = 75$[/tex],
and both values are at least 10.
Thus, all the conditions for inference are satisfied.
──────────────────────────────
Final Conclusion:
The data provide convincing evidence at the [tex]$\alpha=0.01$[/tex] level that greater than [tex]$25\%$[/tex] of adults would describe themselves as organized.
