The task is to apply the Empirical Rule to one of the data sets shown in the histograms (Date 1 and Date 2) for approximations of the temperature distributions. Let's break this down.

The Empirical Rule:

The Empirical Rule applies to normally distributed data, stating that for a normal distribution:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Steps for solving:

(a) Identify the data set for which it is appropriate to use the Empirical Rule:

• Empirical Rule works best for normally distributed data. From the histograms:
  • Date 1 has a more symmetric and bell-shaped distribution, which suggests that it is closer to a normal distribution.
  • Date 2 appears skewed, meaning it is not normally distributed.
• Thus, Date 1 is the appropriate data set to use the Empirical Rule.

(b) Use the Empirical Rule to approximate the percentage of temperatures between 15.68°C and 35.6°C:

• For Date 1:
  • Mean = 31.93°C, Standard Deviation = 7.71°C.
  • 15.68°C and 35.6°C correspond to approximately two standard deviations below and one standard deviation above the mean.
  • Using the Empirical Rule, approximately 81.5% of data falls between one and two standard deviations from the mean.

(c) Approximate 68% of the temperatures for Date 1:

• Since 68% of the data falls within one standard deviation of the mean:
  • The range is $31.93°C \pm 7.71°C$, i.e., between 24.22°C and 39.64°C.

Final Answers:

• (a) The Empirical Rule should be applied to Date 1.
• (b) The percentage of temperatures between 15.68°C and 35.6°C is approximately 81.5%.
• (c) Approximately 68% of the temperatures are between 24.22°C and 39.64°C.

Would you like further clarification, or help with any other parts?

Here are five follow-up questions to expand on this topic:

1. How do you determine if a distribution is normally distributed just by looking at a histogram?
2. What does the Empirical Rule say about data that is three standard deviations from the mean?
3. Why can't we apply the Empirical Rule to skewed data sets?
4. What would happen to the percentage estimates if the standard deviation of a data set increased?
5. Can the Empirical Rule be used for all types of data sets?

Tip: The Empirical Rule is very useful for approximating probabilities for normally distributed data without needing complex calculations.