Questions: Evaluate the formula z=(p̂−p)/√(pq/n) when p̂=110/430, n=430, p=0.25, and q=1-p. z= (Round to two decimal places as needed.)

Transcript text: Evaluate the formula $z=\frac{\hat{p}-p}{\sqrt{\frac{p q}{n}}}$ when $\hat{p}=\frac{110}{430}, n=430, p=0.25$, and $q=1-p$. $z=$ $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate Sample Proportion

The sample proportion $ \hat{p} $ is calculated as follows: \[ \hat{p} = \frac{110}{430} \approx 0.2558 \]

Step 2: Define Parameters

We have the following parameters:

Sample size $ n = 430 $
Hypothesized population proportion $ p = 0.25 $
Complement of the hypothesized proportion $ q = 1 - p = 0.75 $

Step 3: Calculate Standard Error

The standard error (SE) for the proportion is calculated using the formula: \[ SE = \sqrt{\frac{p \cdot q}{n}} = \sqrt{\frac{0.25 \cdot 0.75}{430}} \approx 0.02088 \]

Step 4: Calculate Z-Score

The Z-score is calculated using the formula: \[ z = \frac{\hat{p} - p}{SE} = \frac{0.2558 - 0.25}{0.02088} \approx 0.2784 \]

Step 5: Round Z-Score

Rounding the Z-score to two decimal places gives: \[ z \approx 0.28 \]