Using the Empirieal Rule In Erercises $$ 29-34 $$, use the Empirical Rule. 29. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with astandard deviation of 4 miles per hour. Estimate the percent of velices whose speeds are between 63 miles per hour and 71 miles per hour. (Assume the data set has a bell-shaped distribution.) 30. The mean monthly utility bill for a sample of households in a city is $$ \$ 70 $$, with a standard deviation of $$ \$ 8 $$. Between what two values do about $$ 95 \% $$ of the data lic? (Assume the data set has a bell-shaped distribution.) 31. Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75 . (a) Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (b) In a sample of 25 additional vehicles, about how many vehicles would you expect to have speeds between 63 miles per hour and 71 miles per hour? 32. Use the sample statistics from Exercise 30 and assume the number of households in the sample is 40 . (a) Estimate the number of households whose monthly utility bills are between $$ \$ 54 $$ and $$ \$ 86 $$. (b) In a sample of 20 additional households, about how many households would you expect to have monthly utility bills between $$ \$ 54 $$ and $$ \$ 86 $$ ?

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UpStudy AI · Accepted Answer

Certainly! Let's work through each of the exercises $$29$$ through $$32$$ using the Empirical Rule. The Empirical Rule (also known as the 68-95-99.7 rule) applies to bell-shaped (normally distributed) data sets and helps estimate the percentage of data within certain ranges of the mean.

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## **Exercise 29**

**Problem:**
The mean speed of a sample of vehicles along a stretch of highway is $$67$$ miles per hour, with a standard deviation of $$4$$ miles per hour. Estimate the percent of vehicles whose speeds are between $$63$$ miles per hour and $$71$$ miles per hour. (Assume the data set has a bell-shaped distribution.)

**Solution:**

1. **Identify the Mean and Standard Deviation:**
   - Mean ($$\mu$$) = $$67$$ mph
   - Standard Deviation ($$\sigma$$) = $$4$$ mph

2. **Determine the Range in Terms of $$\sigma$$:**
   - Lower bound = $$63$$ mph
   - Upper bound = $$71$$ mph

3. **Calculate How Many Standard Deviations the Bounds Are from the Mean:**
   - $$63$$ mph = $$\mu - 1\sigma$$ (since $$67 - 4 = 63$$)
   - $$71$$ mph = $$\mu + 1\sigma$$ (since $$67 + 4 = 71$$)

4. **Apply the Empirical Rule:**
   - Approximately **68%** of the data falls within $$\mu \pm 1\sigma$$.

**Answer:**
Approximately **68%** of vehicles have speeds between $$63$$ mph and $$71$$ mph.

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## **Exercise 30**

**Problem:**
The mean monthly utility bill for a sample of households in a city is $$\$70$$, with a standard deviation of $$\$8$$. Between what two values do about $$95\%$$ of the data lie? (Assume the data set has a bell-shaped distribution.)

**Solution:**

1. **Identify the Mean and Standard Deviation:**
   - Mean ($$\mu$$) = $$\$70$$
   - Standard Deviation ($$\sigma$$) = $$\$8$$

2. **Determine the Range for 95% of the Data:**
   - The Empirical Rule states that about **95%** of the data lies within $$\mu \pm 2\sigma$$.

3. **Calculate the Bounds:**
   - Lower bound = $$\mu - 2\sigma = 70 - (2 \times 8) = 70 - 16 = \$54$$
   - Upper bound = $$\mu + 2\sigma = 70 + (2 \times 8) = 70 + 16 = \$86$$

**Answer:**
About **95%** of the monthly utility bills lie between $$\$54$$ and $$\$86$$.

---

## **Exercise 31**

**Problem:**
Use the sample statistics from Exercise $$29$$ and assume the number of vehicles in the sample is $$75$$.

### **Part (a)**

**Estimate the number of vehicles whose speeds are between $$63$$ mph and $$71$$ mph.**

**Solution:**

1. From Exercise $$29$$, we know that approximately **68%** of vehicles fall within $$63$$ mph to $$71$$ mph.

2. **Calculate the Estimated Number of Vehicles:**
   $$
   \text{Number of Vehicles} = 0.68 \times 75 = 51
   $$

**Answer:**
Approximately **51 vehicles** are expected to have speeds between $$63$$ mph and $$71$$ mph.

### **Part (b)**

**In a sample of $$25$$ additional vehicles, about how many vehicles would you expect to have speeds between $$63$$ mph and $$71$$ mph?**

**Solution:**

1. Using the same percentage (**68%**):

2. **Calculate the Expected Number:**
   $$
   \text{Number of Vehicles} = 0.68 \times 25 = 17
   $$

**Answer:**
About **17 vehicles** are expected in the additional sample to have speeds between $$63$$ mph and $$71$$ mph.

---

## **Exercise 32**

**Problem:**
Use the sample statistics from Exercise $$30$$ and assume the number of households in the sample is $$40$$.

### **Part (a)**

**Estimate the number of households whose monthly utility bills are between $$\$54$$ and $$\$86$$.**

**Solution:**

1. From Exercise $$30$$, approximately **95%** of households have bills between $$\$54$$ and $$\$86$$.

2. **Calculate the Estimated Number of Households:**
   $$
   \text{Number of Households} = 0.95 \times 40 = 38
   $$

**Answer:**
Approximately **38 households** are expected to have monthly utility bills between $$\$54$$ and $$\$86$$.

### **Part (b)**

**In a sample of $$20$$ additional households, about how many households would you expect to have monthly utility bills between $$\$54$$ and $$\$86$$?**

**Solution:**

1. Using the same percentage (**95%**):

2. **Calculate the Expected Number:**
   $$
   \text{Number of Households} = 0.95 \times 20 = 19
   $$

**Answer:**
About **19 households** are expected in the additional sample to have monthly utility bills between $$\$54$$ and $$\$86$$.

---

**Summary of Answers:**

- **Exercise 29:** Approximately **68%** of vehicles have speeds between $$63$$ mph and $$71$$ mph.
- **Exercise 30:** About **95%** of the monthly utility bills lie between $$\$54$$ and $$\$86$$.
- **Exercise 31(a):** Approximately **51 vehicles**.
- **Exercise 31(b):** About **17 vehicles**.
- **Exercise 32(a):** Approximately **38 households**.
- **Exercise 32(b):** About **19 households**.

If you have any further questions or need additional explanations, feel free to ask!
**Summary of Answers:** - **Exercise 29:** Approximately **68%** of vehicles have speeds between 63 mph and 71 mph. - **Exercise 30:** About **95%** of the monthly utility bills lie between \$54 and \$86. - **Exercise 31(a):** Approximately **51 vehicles**. - **Exercise 31(b):** About **17 vehicles**. - **Exercise 32(a):** Approximately **38 households**. - **Exercise 32(b):** About **19 households**. If you have any further questions or need additional explanations, feel free to ask!

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