This worksheet is focused on calculating probabilities and values using the normal distribution for a set of lap times with a mean of 1.84 minutes and a standard deviation of 0.07 minutes. I'll go through each question:

Given:

• Mean (μ) = 1.84 minutes
• Standard deviation (σ) = 0.07 minutes

Questions:

1. Label the values 1, 2, and 3 standard deviations above and below the mean.

   • To label these, we add and subtract multiples of the standard deviation from the mean:
     • 1 SD above: $1.84 + 0.07 = 1.91$
     • 2 SD above: $1.84 + 2 \times 0.07 = 1.98$
     • 3 SD above: $1.84 + 3 \times 0.07 = 2.05$
     • 1 SD below: $1.84 - 0.07 = 1.77$
     • 2 SD below: $1.84 - 2 \times 0.07 = 1.70$
     • 3 SD below: $1.84 - 3 \times 0.07 = 1.63$

2. What percentage of the lap times are less than 1.70 minutes?

   • Calculate the z-score: $z = \frac{1.70 - 1.84}{0.07} \approx -2$
   • Using the z-table, a z-score of -2 corresponds to about 2.28%.

3. What percentage of the lap times are greater than 1.77 minutes?

   • Calculate the z-score: $z = \frac{1.77 - 1.84}{0.07} \approx -1$
   • The area to the left of z = -1 is approximately 15.87%, so the percentage greater than 1.77 is: $100\% - 15.87\% = 84.13\%$

4. What percentage of the lap times are between 1.70 and 1.91 minutes?

   • For 1.70, $z \approx -2$ (2.28% below).
   • For 1.91, $z \approx 1$ (84.13% below).
   • The percentage between these two values is: $84.13\% - 2.28\% = 81.85\%$

5. What lap time would be in the slowest 2.5% of all lap times?

   • For the slowest 2.5%, we look for a z-score of about -1.96.
   • Solving for the lap time: $x = \mu + z \cdot \sigma = 1.84 + (-1.96) \cdot 0.07 \approx 1.704$
   • So, 1.704 minutes.

6. What proportion of laps were less than 1.72 minutes?

   • Calculate the z-score: $z = \frac{1.72 - 1.84}{0.07} \approx -1.71$
   • A z-score of -1.71 corresponds to approximately 4.38%, so the proportion is 4.38%.

Would you like any further explanations on these calculations?

Here are five related questions to expand your understanding:

1. How would the percentages change if the standard deviation were larger?
2. How can we interpret z-scores and their corresponding areas?
3. Why is it that approximately 68% of data falls within one standard deviation in a normal distribution?
4. How can we calculate probabilities for values greater than three standard deviations from the mean?
5. What happens if the distribution is not normal?

Tip: When working with z-scores, remember that they represent the number of standard deviations a value is from the mean. This makes it easier to compare values across different distributions.