Step-by-step explanation:
To find the percentage of participants whose satisfaction rating falls between 4.75 and 6.75 seconds, we can use the z-score formula and then look up the corresponding percentages in a standard normal distribution table.
First, let's find the z-scores for both 4.75 and 6.75 seconds using the formula:
\[ z = \frac{{X - \mu}}{{\sigma}} \]
Where:
- \( X \) = individual satisfaction rating (4.75 and 6.75 seconds)
- \( \mu \) = mean satisfaction rating (5.5 seconds)
- \( \sigma \) = standard deviation (0.5 seconds)
For 4.75 seconds:
\[ z_1 = \frac{{4.75 - 5.5}}{{0.5}} = -1 \]
For 6.75 seconds:
\[ z_2 = \frac{{6.75 - 5.5}}{{0.5}} = 2.5 \]
Now, we can look up the corresponding percentages in the standard normal distribution table. The percentage between these two z-scores represents the percentage of participants whose satisfaction rating falls between 4.75 and 6.75 seconds. From the table, we find:
- For \( z_1 = -1 \), the percentage is approximately 15.87%
- For \( z_2 = 2.5 \), the percentage is approximately 99.38%
To find the percentage between the two z-scores, we subtract the percentage corresponding to the lower z-score from the percentage corresponding to the higher z-score:
\[ \text{Percentage} = 99.38\% - 15.87\% = 83.51\% \]
So, approximately 83.51% of the participants will have a satisfaction rating between 4.75 and 6.75 seconds.
