Questions: Internet service: An Internet service provider sampled 536 customers, and finds that 79 of them experienced an interruption in high-speed service during the previous month. Part 1 of 3 (a) Find a point estimate for the population proportion of all customers who experienced an interruption. Round the answer to at least three decimal places. The point estimate for the population proportion of all customers who experienced an interruption is 0.147. Part 2 of 3 (b) Construct a 95% confidence interval for the proportion of all customers who experienced an interruption. Round the answers to at least three decimal places.

Solution Steps

To find the point estimate for the population proportion of all customers who experienced an interruption in high-speed service, we use the formula:

\[ \hat{p} = \frac{x}{n} \]

where:

• \( x = 79 \) (the number of customers who experienced an interruption)
• \( n = 536 \) (the total number of customers sampled)

Calculating this gives:

\[ \hat{p} = \frac{79}{536} \approx 0.147 \]

Thus, the point estimate for the population proportion is:

\[ \hat{p} \approx 0.147 \]

To construct a \( 95\% \) confidence interval for the population proportion, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

For a \( 95\% \) confidence level, the critical value \( z \) is approximately \( 1.96 \).

Substituting the values:

\[ \hat{p} = 0.147, \quad n = 536 \]

Calculating the standard error:

\[ \text{SE} = \sqrt{\frac{0.147(1 - 0.147)}{536}} \approx \sqrt{\frac{0.147 \cdot 0.853}{536}} \approx 0.015 \]

Now, we can calculate the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \text{SE} \approx 1.96 \cdot 0.015 \approx 0.0294 \]

Thus, the confidence interval is:

\[ 0.147 \pm 0.0294 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 0.147 - 0.0294 \approx 0.117 \] \[ \text{Upper Bound} = 0.147 + 0.0294 \approx 0.177 \]

Therefore, the \( 95\% \) confidence interval for the proportion of all customers who experienced an interruption is:

\[ 0.117 < p < 0.177 \]

Final Answer