https://file.aiquickdraw.com/mathbot/file/173044967957165hdi7b6.jpg

To answer the questions in the image, we’ll calculate the ranges for 2017 and 2021 using their mean temperatures and standard deviations. Let's go through each question.

### Question (a): Finding the 95% Range for 2017 and 2021

For a normal distribution, 95% of the data falls within **two standard deviations** from the mean. Thus, the range can be calculated by adding and subtracting twice the standard deviation from the mean.

The values provided in the table are:
- **2017 Mean:** 83.44, **Standard Deviation:** 0.83
- **2021 Mean:** 81.12, **Standard Deviation:** 0.71
- **2023 Mean:** 81.41, **Standard Deviation:** 0.60 (given in the question as 80.19 to 82.61 for 2023)

#### Step 1: Calculate the 95% Range for 2017
\[
\text{Range for 2017} = \text{Mean} \pm 2 \times \text{Standard Deviation}
\]
\[
= 83.44 \pm 2 \times 0.83
\]
\[
= 83.44 \pm 1.66
\]
\[
= [83.44 - 1.66, 83.44 + 1.66]
\]
\[
= [81.78, 85.10]
\]
So, the **95% range for 2017 is approximately 81.78°F to 85.10°F**.

#### Step 2: Calculate the 95% Range for 2021
\[
\text{Range for 2021} = \text{Mean} \pm 2 \times \text{Standard Deviation}
\]
\[
= 81.12 \pm 2 \times 0.71
\]
\[
= 81.12 \pm 1.42
\]
\[
= [81.12 - 1.42, 81.12 + 1.42]
\]
\[
= [79.70, 82.54]
\]
So, the **95% range for 2021 is approximately 79.70°F to 82.54°F**.

### Summary for Part (a)
- **2017 Range (95%):** 81.78°F to 85.10°F
- **2021 Range (95%):** 79.70°F to 82.54°F
- **2023 Range (95%)** (provided): 80.19°F to 82.61°F

---

### Question (b): Comparing Mean Temperatures Between 2023 and 2017

1. **Absolute Difference**: The absolute difference in mean temperature between 2023 and 2017 can be found as follows:
   \[
   \text{Absolute Difference} = 83.44 - 81.41 = 2.03 \text{°F}
   \]
   So, **2023’s mean temperature is 2.03°F lower than 2017’s mean**.

2. **Relative Difference (Percentage)**: To express this difference as a percentage of 2017’s mean, we use the formula:
   \[
   \text{Relative Difference} = \left(\frac{2.03}{83.44}\right) \times 100\%
   \]
   \[
   = 2.43\%
   \]
   Therefore, the **mean temperature in 2023 is approximately 2.43% lower than in 2017**.

### Summary for Part (b)
- **Absolute Difference:** 2.03°F
- **Relative Percentage Difference:** 2.43%

Would you like further clarification on any of these calculations or additional insights?

### Additional Related Questions
1. What could cause fluctuations in temperature over these years?
2. How does the temperature distribution in 2023 compare visually with that of 2017?
3. How would the 2023 temperature range change if it had the same standard deviation as 2017?
4. Why is it helpful to use both absolute and relative differences in comparisons?
5. How would the temperature distributions shift if climate changes caused consistent warming?

### Tip
For temperature comparisons, absolute differences give a clear picture, but relative percentages allow you to gauge significance, especially when comparing values of different scales.

Based on the temperature data provided in the image, calculate the 95% range for temperatures in 2017 and 2021, and compare the mean temperature of 2023 with 2017 in terms of absolute and relative percentage differences.

Explore the calculation of temperature ranges for 2017 and 2021 using standard deviation, and compare the mean temperatures of 2023 and 2017. Learn how to determine the 95% range in a normal distribution and assess both absolute and relative differences in means. Suitable for high school statistics.

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