To address the questions in the image, let’s break down the requirements for each part based on the empirical rule (also known as the 68-95-99.7 rule), which applies to data that is approximately normally distributed. The empirical rule states:

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

Analysis of the Data Sets

1. Date 1:

   • Mean: $23.96^\circ C$
   • Standard Deviation: $4.58^\circ C$

   The histogram for Date 1 appears more symmetric and resembles a normal distribution.

2. Date 2:

   • Mean: $24.65^\circ C$
   • Standard Deviation: $10.21^\circ C$

   The histogram for Date 2 is less symmetric and has a broader spread, suggesting it may not follow a normal distribution as closely as Date 1.

Given this information, the empirical rule is more appropriate to use with Date 1, as it more closely resembles a normal distribution.

Answers to the Questions

Part (a)

• Identify the data set for which it is appropriate to use the Empirical Rule: The appropriate data set is Date 1.

Part (b)

• Percentage of temperatures between 10.22°C and 37.7°C for Date 1:
  • We calculate how many standard deviations these values are from the mean for Date 1.
    • Mean = $23.96^\circ C$
    • Standard Deviation = $4.58^\circ C$
    • Lower Bound: $10.22^\circ C$ $(\approx 3 \text{ standard deviations below the mean})$
    • Upper Bound: $37.7^\circ C$ $(\approx 3 \text{ standard deviations above the mean})$
  • Therefore, approximately 99.7% of temperatures fall within this range according to the empirical rule.

Part (c)

• Approximate temperature range covering 95% of the data for Date 1:
  • 95% of the data falls within 2 standard deviations of the mean.
    • Lower Bound = Mean - 2 × Standard Deviation = $23.96 - 2 \times 4.58 = 14.8^\circ C$
    • Upper Bound = Mean + 2 × Standard Deviation = $23.96 + 2 \times 4.58 = 33.12^\circ C$
  • So, approximately 95% of the temperatures are between 14.8°C and 33.12°C.

Would you like more details on any part or have further questions?

Expanding Questions:

1. How can we determine if a data set is approximately normal based on its histogram?
2. What is the difference between using the empirical rule and calculating exact probabilities?
3. How can we approximate values that fall outside the normal distribution?
4. Why might the empirical rule not apply to Date 2's data set?
5. How would skewness in data affect the applicability of the empirical rule?

Tip:

When analyzing data, always check the shape of the distribution before applying the empirical rule to ensure it’s suitable for approximation.