SOLUTION: Help needed please.... Consider the following hypotheses: H0: p ≥ 0.47 HA: p < 0.47 Compute the p-value based on the following sample information. (You may find

Question 1163125: Help needed please....
Consider the following hypotheses:
H0: p ≥ 0.47
HA: p < 0.47
Compute the p-value based on the following sample information. (You may find it useful to reference the appropriate table: z table or t table) (Round "z" value to 2 decimal places. Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)
p-value
a. x = 46; n = 118 _______
b. x = 112; n = 311 _______
c. p¯ = 0.38; n = 53 _______
d. p¯ = 0.38; n = 430 _______
Answer by CPhill(2264) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import math
from scipy.stats import norm
def calculate_p_value(n, p0, x=None, p_hat=None):
if p_hat is None:
p_hat = x / n

# Standard Error
se = math.sqrt(p0 * (1 - p0) / n)

# Test statistic z
z = (p_hat - p0) / se

# Round z to 2 decimal places as per instructions
z_rounded = round(z, 2)

# Lower-tailed p-value
p_val = norm.cdf(z_rounded)

return {
"p_hat": round(p_hat, 6),
"se": round(se, 6),
"z": round(z, 6),
"z_rounded": z_rounded,
"p_value": round(p_val, 4)
}
p0 = 0.47
results = {
"a": calculate_p_value(n=118, p0=p0, x=46),
"b": calculate_p_value(n=311, p0=p0, x=112),
"c": calculate_p_value(n=53, p0=p0, p_hat=0.38),
"d": calculate_p_value(n=430, p0=p0, p_hat=0.38)
}
import json
print(json.dumps(results, indent=2))

```
```text?code_stdout&code_event_index=2
{
"a": {
"p_hat": 0.389831,
"se": 0.045946,
"z": -1.744871,
"z_rounded": -1.74,
"p_value": 0.0409
},
"b": {
"p_hat": 0.360129,
"se": 0.028301,
"z": -3.8822,
"z_rounded": -3.88,
"p_value": 0.0001
},
"c": {
"p_hat": 0.38,
"se": 0.068557,
"z": -1.312785,
"z_rounded": -1.31,
"p_value": 0.0951
},
"d": {
"p_hat": 0.38,
"se": 0.024069,
"z": -3.739296,
"z_rounded": -3.74,
"p_value": 0.0001
}
}
```
To compute the p-values for these lower-tailed hypothesis tests, we use the following steps:
1. **Calculate the sample proportion ($\hat{p}$):** $\hat{p} = \frac{x}{n}$ (if not already given).
2. **Calculate the standard error ($SE$):** $SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$ where $p_0 = 0.47$.
3. **Calculate the test statistic ($z$):** $z = \frac{\hat{p} - p_0}{SE}$.
4. **Find the p-value:** Since $H_A: p < 0.47$, the p-value is $P(Z \leq z_{rounded})$.
Here are the step-by-step calculations for each part:
### a. $x = 46$; $n = 118$
* **Sample Proportion ($\hat{p}$):** $46 / 118 \approx 0.3898$
* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{118}} \approx 0.0459$
* **Test Statistic ($z$):** $\frac{0.3898 - 0.47}{0.0459} \approx -1.74$
* **p-value:** $P(Z \leq -1.74) = \mathbf{0.0409}$
### b. $x = 112$; $n = 311$
* **Sample Proportion ($\hat{p}$):** $112 / 311 \approx 0.3601$
* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{311}} \approx 0.0283$
* **Test Statistic ($z$):** $\frac{0.3601 - 0.47}{0.0283} \approx -3.88$
* **p-value:** $P(Z \leq -3.88) = \mathbf{0.0001}$ (approx.)
### c. $\bar{p} = 0.38$; $n = 53$
* **Sample Proportion ($\hat{p}$):** $0.38$
* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{53}} \approx 0.0686$
* **Test Statistic ($z$):** $\frac{0.38 - 0.47}{0.0686} \approx -1.31$
* **p-value:** $P(Z \leq -1.31) = \mathbf{0.0951}$
### d. $\bar{p} = 0.38$; $n = 430$
* **Sample Proportion ($\hat{p}$):** $0.38$
* **Standard Error ($SE$):** $\sqrt{\frac{0.47 \times 0.53}{430}} \approx 0.0241$
* **Test Statistic ($z$):** $\frac{0.38 - 0.47}{0.0241} \approx -3.74$
* **p-value:** $P(Z \leq -3.74) = \mathbf{0.0001}$ (approx.)
---
**Final Answers:**
* a. **0.0409**
* b. **0.0001**
* c. **0.0951**
* d. **0.0001**
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