 Bob runs a phone company called Phantastic Phones. He conducts a survey on a random sample of 500 of his customers, in which he asks questions about their phone use habits, expenditure on phones, and so on.



The survey showed that the sample proportion of customers who had a pre-paid plan was 0.64.  Calculate the standard error associated with this estimate (correct to 4 dp)?



To calculate the standard error (SE) of the sample proportion, we use the formula:

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

Where:
- \( p \) is the sample proportion (0.64 in this case),
- \( n \) is the sample size (500).

### Step-by-step calculation:
1. **Substitute the given values** into the formula:
   \[
   SE = \sqrt{\frac{0.64(1 - 0.64)}{500}} = \sqrt{\frac{0.64 \times 0.36}{500}}
   \]
   
2. **Simplify the numerator**:
   \[
   0.64 \times 0.36 = 0.2304
   \]
   
3. **Divide by the sample size**:
   \[
   \frac{0.2304}{500} = 0.0004608
   \]

4. **Take the square root**:
   \[
   \sqrt{0.0004608} \approx 0.0215
   \]

### Final Answer:
The standard error (SE) is approximately **0.0215** (correct to 4 decimal places).

Let me know if you'd like more details or have any questions!

Here are some related questions:
1. How does the sample size affect the standard error?
2. How can we calculate the margin of error for this survey result?
3. What does the standard error represent in the context of this survey?
4. How would the standard error change if the sample proportion were 0.50 instead of 0.64?
5. How can we use the standard error to construct a confidence interval for the population proportion?

Tip: Increasing the sample size generally reduces the standard error, leading to more accurate estimates of population parameters.

The survey showed that the sample proportion of customers who had a pre-paid plan was 0.64. Calculate the standard error associated with this estimate (correct to 4 dp).

Learn how to calculate the standard error for a sample proportion in a survey of pre-paid phone plans. In this case, with a sample size of 500 and a proportion of 0.64, the standard error is 0.0215. Detailed step-by-step calculation and explanation included.

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