A random sample of \( n=755 \) us cell phone users age 18 and older in May 2011 found that the average number of text messages sent or received per day is 41.5 messages, \( { }^{1} \) with standard error about 6.1 . (b) Find a \( 95 \% \) confidence interval for the mean number of text messages. The \( 95 \% \) confidence interval is

To calculate the \( 95 \% \) confidence interval for the mean number of text messages, we can use the formula: \[ \text{Confidence Interval} = \bar{x} \pm z^{*} \cdot \frac{s}{\sqrt{n}} \] Where: - \( \bar{x} = 41.5 \) (sample mean) - \( z^{*} \) is the z-score corresponding to the confidence level (for \( 95 \% \), \( z^{*} \approx 1.96 \)) - \( s = 6.1 \) (standard error) - \( n = 755 \) (sample size) First, we calculate the standard error: \[ \text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{6.1}{\sqrt{755}} \approx 0.221 \] Now, we calculate the margin of error: \[ \text{Margin of Error} = z^{*} \cdot \text{Standard Error} \approx 1.96 \cdot 0.221 \approx 0.433 \] Finally, we can find the confidence interval: \[ \text{Confidence Interval} = 41.5 \pm 0.433 \] Calculating the lower and upper limits: \[ \text{Lower limit} = 41.5 - 0.433 \approx 41.067 \] \[ \text{Upper limit} = 41.5 + 0.433 \approx 41.933 \] Thus, the \( 95 \% \) confidence interval is approximately \( (41.07, 41.93) \).