Question

the graph shows the distribution of the number of text messages young adults send per day. the distribution is approximately normal, with a mean of 128 messages and a standard deviation of 30 messages. what percentage of young adults send between 128 and 158 text messages per day? 34% 50% 66% 68%

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately 68% of the data lies within \( \mu\pm\sigma\) (where \(\mu\) is the mean and \(\sigma\) is the standard deviation)
• Approximately 95% of the data lies within \( \mu\pm2\sigma\)
• Approximately 99.7% of the data lies within \( \mu\pm3\sigma\)

Step2: Calculate the range in terms of mean and standard deviation

Given \(\mu = 128\) (mean) and \(\sigma=30\) (standard deviation)
The lower - bound of the range is \(128\) (\(\mu\)) and the upper - bound is \(128 + 30=158\) (\(\mu+\sigma\))
According to the empirical rule, the percentage of data within \(\mu\) and \(\mu + \sigma\) is \(34\%\)

Answer:

34%