For the numbers below, find the percent between the mean and the \( z \)-scor a) \( z=1.17 \) b) \( z=-1.37 \)

To find the percent between the mean and the \( z \)-score, we first need to understand what a \( z \)-score represents. A \( z \)-score indicates how many standard deviations a data point is from the mean. The area under the standard normal distribution curve can be used to find the percentage of values that fall below a given \( z \)-score. We can use a standard normal distribution table (or a calculator) to find the cumulative probability associated with each \( z \)-score. ### a) For \( z = 1.17 \) 1. Look up the cumulative probability for \( z = 1.17 \): - From the standard normal distribution table, the cumulative probability for \( z = 1.17 \) is approximately \( 0.8790 \). 2. To find the percent between the mean (which corresponds to \( z = 0 \)) and \( z = 1.17 \): - The cumulative probability at the mean (z = 0) is \( 0.5 \). - The area between the mean and \( z = 1.17 \) is: \[ P(0 < z < 1.17) = P(z < 1.17) - P(z < 0) = 0.8790 - 0.5 = 0.3790 \] 3. Convert this to a percentage: \[ 0.3790 \times 100 = 37.90\% \] ### b) For \( z = -1.37 \) 1. Look up the cumulative probability for \( z = -1.37 \): - From the standard normal distribution table, the cumulative probability for \( z = -1.37 \) is approximately \( 0.0853 \). 2. To find the percent between the mean and \( z = -1.37 \): - The cumulative probability at the mean (z = 0) is \( 0.5 \). - The area between the mean and \( z = -1.37 \) is: \[ P(-1.37 < z < 0) = P(z < 0) - P(z < -1.37) = 0.5 - 0.0853 = 0.4147 \] 3. Convert this to a percentage: \[ 0.4147 \times 100 = 41.47\% \] ### Summary: - For \( z = 1.17 \), the percent between the mean and the \( z \)-score is **37.90%**. - For \( z = -1.37 \), the percent between the mean and the \( z \)-score is **41.47%**.