Question

mean=1.84 standard deviation=0.07 9. what lap time is necessary for mario to be in the fastest 3% of all his laps? (remember a fast lap time is a low lap time)

Explanation:

Step1: Identify the z-score for the fastest 3%

We need to find the z-score corresponding to the lower 3% (since fast lap times are low, we look at the left tail of the normal distribution). Using a z-table or calculator, the z-score for which \( P(Z < z) = 0.03 \) is approximately \( z = -1.88 \) (from standard normal distribution tables or inverse normal function).

Step2: Use the z-score formula to find the lap time

The z-score formula is \( z=\frac{x - \mu}{\sigma} \), where \( \mu = 1.84 \) (mean), \( \sigma = 0.07 \) (standard deviation), and \( z=-1.88 \). We solve for \( x \):

\[
x=\mu + z\times\sigma
\]

Substitute the values:

\[
x = 1.84+(-1.88)\times0.07
\]

\[
x = 1.84 - 0.1316
\]

\[
x = 1.7084
\]

Answer:

The necessary lap time for Mario to be in the fastest 3% of all his laps is approximately \( 1.71 \) (or more precisely \( 1.7084 \)).