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 more than 158 text messages per day? 16% 34% 68% 84%

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x - \mu}{\sigma}\), where \(x = 158\), \(\mu=128\), and \(\sigma = 30\).

$$z=\frac{158 - 128}{30}=\frac{30}{30}=1$$

Step2: Use the empirical rule

The empirical rule for a normal distribution states that about \(68\%\) of the data lies within \(z=- 1\) and \(z = 1\), about \(95\%\) lies within \(z=-2\) and \(z = 2\), and about \(99.7\%\) lies within \(z=-3\) and \(z = 3\).
The area to the left of \(z = 1\) is \(0.84\) (from the standard normal distribution table or the empirical rule: \(50\%+34\% = 84\%\) for the area to the left of \(z = 1\)).
The area to the right of \(z = 1\) is \(1-0.84=0.16\)

Answer:

16%