Answer :
We wish to test whether more than 25% of adults describe themselves as organized. Let
[tex]$$
H_0: p = 0.25 \quad \text{and} \quad H_a: p > 0.25.
$$[/tex]
Below is a step-by-step solution.
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Step 1. Verify Conditions for Inference
1. Random Sampling:
A random sample of 100 adults was taken.
2. 10% Condition:
The sample size (100) is less than 10% of the population of all adults. This condition is satisfied.
3. Large Counts Condition:
Under [tex]$H_0$[/tex], the expected number of adults describing themselves as organized is
[tex]$$
np_0 = 100 \times 0.25 = 25,
$$[/tex]
and the expected number not describing themselves as organized is
[tex]$$
n(1-p_0) = 100 \times 0.75 = 75.
$$[/tex]
Both values (25 and 75) are at least 10.
Thus, all conditions for inference are met.
──────────────────────────────
Step 2. Compute the Test Statistic
1. Sample Proportion:
Given 42 out of 100 adults describe themselves as organized, we have
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
2. Standard Error:
The standard error (using [tex]$p_0$[/tex]) is calculated as
[tex]$$
SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{100}} \approx 0.04330.
$$[/tex]
3. Z-Statistic:
The test statistic is then
[tex]$$
z = \frac{\hat{p} - p_0}{SE} = \frac{0.42 - 0.25}{0.04330} \approx 3.93.
$$[/tex]
──────────────────────────────
Step 3. Determine the Critical Value and P-value
1. Critical Value:
For a one-tailed test at a significance level [tex]$\alpha = 0.01$[/tex], the critical value is
[tex]$$
z_{\text{crit}} \approx 2.33.
$$[/tex]
2. P-value:
The p-value corresponding to [tex]$z \approx 3.93$[/tex] is approximately
[tex]$$
p\text{-value} \approx 4.32 \times 10^{-5}.
$$[/tex]
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Step 4. Decision
Since the test statistic [tex]$z \approx 3.93$[/tex] exceeds the critical value [tex]$2.33$[/tex], and the p-value is far less than [tex]$\alpha = 0.01$[/tex], we reject the null hypothesis [tex]$H_0$[/tex].
──────────────────────────────
Conclusion
There is convincing evidence at the [tex]$\alpha = 0.01$[/tex] significance level that more than 25% of adults would describe themselves as organized.
──────────────────────────────
Summary of Conditions
- Random: We have a random sample of 100 adults.
- 10% Condition: 100 is less than 10% of the population of adults.
- Large Counts:
[tex]$$ np_0 = 25 \quad \text{and} \quad n(1-p_0) = 75, $$[/tex]
which are both at least 10.
This completes the detailed step-by-step solution.
[tex]$$
H_0: p = 0.25 \quad \text{and} \quad H_a: p > 0.25.
$$[/tex]
Below is a step-by-step solution.
──────────────────────────────
Step 1. Verify Conditions for Inference
1. Random Sampling:
A random sample of 100 adults was taken.
2. 10% Condition:
The sample size (100) is less than 10% of the population of all adults. This condition is satisfied.
3. Large Counts Condition:
Under [tex]$H_0$[/tex], the expected number of adults describing themselves as organized is
[tex]$$
np_0 = 100 \times 0.25 = 25,
$$[/tex]
and the expected number not describing themselves as organized is
[tex]$$
n(1-p_0) = 100 \times 0.75 = 75.
$$[/tex]
Both values (25 and 75) are at least 10.
Thus, all conditions for inference are met.
──────────────────────────────
Step 2. Compute the Test Statistic
1. Sample Proportion:
Given 42 out of 100 adults describe themselves as organized, we have
[tex]$$
\hat{p} = \frac{42}{100} = 0.42.
$$[/tex]
2. Standard Error:
The standard error (using [tex]$p_0$[/tex]) is calculated as
[tex]$$
SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{100}} \approx 0.04330.
$$[/tex]
3. Z-Statistic:
The test statistic is then
[tex]$$
z = \frac{\hat{p} - p_0}{SE} = \frac{0.42 - 0.25}{0.04330} \approx 3.93.
$$[/tex]
──────────────────────────────
Step 3. Determine the Critical Value and P-value
1. Critical Value:
For a one-tailed test at a significance level [tex]$\alpha = 0.01$[/tex], the critical value is
[tex]$$
z_{\text{crit}} \approx 2.33.
$$[/tex]
2. P-value:
The p-value corresponding to [tex]$z \approx 3.93$[/tex] is approximately
[tex]$$
p\text{-value} \approx 4.32 \times 10^{-5}.
$$[/tex]
──────────────────────────────
Step 4. Decision
Since the test statistic [tex]$z \approx 3.93$[/tex] exceeds the critical value [tex]$2.33$[/tex], and the p-value is far less than [tex]$\alpha = 0.01$[/tex], we reject the null hypothesis [tex]$H_0$[/tex].
──────────────────────────────
Conclusion
There is convincing evidence at the [tex]$\alpha = 0.01$[/tex] significance level that more than 25% of adults would describe themselves as organized.
──────────────────────────────
Summary of Conditions
- Random: We have a random sample of 100 adults.
- 10% Condition: 100 is less than 10% of the population of adults.
- Large Counts:
[tex]$$ np_0 = 25 \quad \text{and} \quad n(1-p_0) = 75, $$[/tex]
which are both at least 10.
This completes the detailed step-by-step solution.
