Suppose that 'p' represents the population proportion of arid climate-zone regions that has a severe pollution level, and suppose that someone claims that p has the value of 23%.

Round all answers to 4 decimal places; do not use intermediate rounding.

a) What would be the Wald standard error associated to using your sample proportion as an estimate for 'p'? the sample proportion is 11/61 [a]

b) What would be the positive/right critical value associated to a symmetric two-tailed alpha value of 18.7%? [b]

c) What would be the associated margin of error for a 81.3% confidence interval? [c]

d) The associated 81.3% Wald confidence interval will be ([d1], [d2]).

e) What would be the associated standard test statistic be equal to? [e]

f) What would be the associated p-value to the test statistic in (e)? [f]

Let's go through each part of the question step-by-step and calculate the values:

### **Given Data:**
- Claimed population proportion \( p = 0.23 \)
- Sample proportion \( \hat{p} = \frac{11}{61} \)
- Sample size \( n = 61 \)
- Alpha level \( \alpha = 0.187 \) for a symmetric two-tailed test

### **(a) Wald Standard Error** 

The Wald standard error is calculated as:

\[
\text{Wald SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( \hat{p} = \frac{11}{61} \)
- \( n = 61 \)

Substitute values:

\[
\text{Wald SE} = \sqrt{\frac{\frac{11}{61}(1 - \frac{11}{61})}{61}}
\]

Let's calculate the Wald standard error.

---

### **(b) Critical Value for a Symmetric Two-Tailed Test**

Given that \( \alpha = 0.187 \), we need to find the critical value for a two-tailed test. This means:

\[
\text{Critical region} = \frac{\alpha}{2} = 0.0935
\]

Using the standard normal (Z) distribution, the positive critical value corresponds to the Z-value at \( 1 - 0.0935 = 0.9065 \).

We can find the corresponding Z-score for this probability using the Z-distribution table or a calculator.

---

### **(c) Margin of Error**

The margin of error (ME) for a confidence interval is given by:

\[
\text{ME} = Z_{\alpha/2} \times \text{Wald SE}
\]

Where:
- \( Z_{\alpha/2} \) is the critical value from part (b)
- \( \text{Wald SE} \) is the standard error from part (a)

Let's calculate the margin of error once we have the critical value.

---

### **(d) Wald Confidence Interval**

The confidence interval is calculated as:

\[
\text{CI} = \left( \hat{p} - \text{ME}, \hat{p} + \text{ME} \right)
\]

We'll substitute the values of \( \hat{p} \) and ME from previous parts to find the confidence interval.

---

### **(e) Standard Test Statistic**

The standard test statistic (Z) is given by:

\[
Z = \frac{\hat{p} - p}{\text{Wald SE}}
\]

Where:
- \( \hat{p} \) is the sample proportion
- \( p = 0.23 \) is the claimed population proportion
- \( \text{Wald SE} \) is the standard error calculated in part (a)

We'll calculate this after computing the standard error.

---

### **(f) P-value**

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed. For a two-tailed test, the p-value is:

\[
\text{P-value} = 2 \times P(Z \geq |Z_{\text{test}}|)
\]

We'll use the standard normal table to find this probability.

---

Let’s calculate each of these steps one by one. I’ll start with the calculations.Here are the calculated results:

### (a) Wald Standard Error:
\[
\text{Wald SE} = 0.0492
\]

### (b) Critical Value for Two-Tailed Test (Alpha = 18.7%):
\[
Z_{\alpha/2} = 1.3195
\]

### (c) Margin of Error:
\[
\text{ME} = 0.0650
\]

### (d) Wald Confidence Interval:
\[
\text{CI} = \left( 0.1154, 0.2453 \right)
\]

### (e) Standard Test Statistic:
\[
Z = -1.0091
\]

### (f) P-value:
\[
\text{P-value} = 0.3129
\]

Let me know if you'd like more details on any part of the calculations or the interpretation.

Here are 5 related questions:
1. How would the confidence interval change with a different sample size?
2. What would the impact be on the margin of error if the confidence level was increased?
3. How is the critical value related to the alpha level in a hypothesis test?
4. How can we interpret the p-value in the context of hypothesis testing?
5. What are the limitations of the Wald confidence interval method?

**Tip**: In hypothesis testing, a higher p-value suggests less evidence against the null hypothesis. Always compare the p-value to your significance level to make conclusions.

Suppose that 'p' represents the population proportion of arid climate-zone regions that has a severe pollution level, and suppose that someone claims that p has the value of 23%. Round all answers to 4 decimal places; do not use intermediate rounding. a) What would be the Wald standard error associated to using your sample proportion as an estimate for 'p'? the sample proportion is 11/61. b) What would be the positive/right critical value associated to a symmetric two-tailed alpha value of 18.7%? c) What would be the associated margin of error for a 81.3% confidence interval? d) The associated 81.3% Wald confidence interval will be ([d1], [d2]). e) What would be the associated standard test statistic be equal to? f) What would be the associated p-value to the test statistic in (e)?

This problem involves estimating a population proportion (p) for pollution levels in arid climate zones, using hypothesis testing, confidence intervals, and Wald standard error calculations. Key steps include calculating the margin of error for a two-tailed test at alpha = 18.7% and finding the associated p-value for the test statistic.

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