Step-by-step explanation:
z = \frac{{X - \mu}}{{\sigma}}
To find the percentage of participants whose satisfaction rating falls between 5.25 and 5.75 seconds, you can use the z-score formula and then look up the corresponding probabilities in a standard normal distribution table.
First, calculate the z-scores for both 5.25 and 5.75 seconds using the formula:
\[ z = \frac{{X - \mu}}{{\sigma}} \]
Where:
- \( X \) is the individual satisfaction rating (either 5.25 or 5.75)
- \( \mu \) is the mean (average) satisfaction rating (5.5 seconds)
- \( \sigma \) is the standard deviation (0.5 seconds)
For 5.25 seconds:
\[ z_{5.25} = \frac{{5.25 - 5.5}}{{0.5}} = -0.5 \]
For 5.75 seconds:
\[ z_{5.75} = \frac{{5.75 - 5.5}}{{0.5}} = 0.5 \]
Next, look up the probabilities corresponding to these z-scores in the standard normal distribution table. The area between -0.5 and 0.5 represents the percentage of participants whose satisfaction ratings fall between 5.25 and 5.75 seconds.
Using a standard normal distribution table or calculator, you find that the area between -0.5 and 0.5 is approximately 0.3829.
So, approximately 38.29% of participants will have satisfaction ratings between 5.25 and 5.75 seconds.
