Questions: About what percent of x values lie between the first and third standard deviations (both sides)? (Enter an exact number as an integer, fraction, or decimal.)

$About what percent of x values lie between the first and third standard deviations (both sides)? (Enter an exact number as an integer, fraction, or decimal.)$

Transcript text: About what percent of $x$ values lie between the first and third standard deviations (both sides)? (Enter an exact.number as an integer, fraction, or decimal.)

Solution

Solution Steps

Step 1: Calculate Z-scores

To determine the percentage of $ x $ values that lie between the first and third standard deviations, we first calculate the Z-scores for the lower and upper bounds. The Z-scores are given by:

\[ Z_{start} = -3.0, \quad Z_{end} = 3.0 \]

Step 2: Calculate Probability

Next, we calculate the probability that a value falls between these Z-scores using the cumulative distribution function $ \Phi $:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(3.0) - \Phi(-3.0) \]

From the output, we find:

\[ P = 0.9973 \]

Step 3: Convert Probability to Percentage

To express this probability as a percentage, we multiply by 100:

\[ \text{Percentage} = P \times 100 = 0.9973 \times 100 = 99.72999999999999\% \]