The mean travel time to work for individuals in a
particular city is = 31.5 minutes. The population
standard deviation o is = 12 minutes. A sample of
50 residents of the city is selected.
(i) Show the sampling distribution of x where x is the sample mean travel time to work for the 50 residents.
(ii) What is the probability that the sample mean will be within # 1 minute of the
population mean?
(iti) What is the probability that the sample mean will be within 3 minutes of the population mean? Calculate properly n explain also

To solve the problem, we analyze the sampling distribution and calculate the probabilities. Let’s break it down step by step.

---

### **(i) Sampling Distribution of \( \bar{x} \)**

The sampling distribution of the sample mean \( \bar{x} \) has the following properties:

1. **Mean (\( \mu_{\bar{x}} \)):**  
   The mean of the sampling distribution is equal to the population mean:  
   \[
   \mu_{\bar{x}} = \mu = 31.5 \, \text{minutes}.
   \]

2. **Standard Error (\( \sigma_{\bar{x}} \)):**  
   The standard error of the sample mean is calculated using the formula:  
   \[
   \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{50}} = \frac{12}{7.071} \approx 1.697 \, \text{minutes}.
   \]

Thus, the sampling distribution of \( \bar{x} \) is normally distributed with:  
\[
\bar{x} \sim N(31.5, 1.697).
\]

---

### **(ii) Probability that \( \bar{x} \) is within 1 minute of the population mean**

We are asked for the probability that \( \bar{x} \) is within 1 minute of the population mean, i.e.,  
\[
P(30.5 \leq \bar{x} \leq 32.5).
\]

We convert the values to standard \( z \)-scores using the formula:  
\[
z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}.
\]

For \( \bar{x} = 30.5 \):  
\[
z = \frac{30.5 - 31.5}{1.697} = \frac{-1}{1.697} \approx -0.589.
\]

For \( \bar{x} = 32.5 \):  
\[
z = \frac{32.5 - 31.5}{1.697} = \frac{1}{1.697} \approx 0.589.
\]

The probability is:  
\[
P(30.5 \leq \bar{x} \leq 32.5) = P(-0.589 \leq z \leq 0.589).
\]

Using standard normal distribution tables or a calculator, we find:  
\[
P(-0.589 \leq z \leq 0.589) \approx 0.4433.
\]

Thus, the probability is approximately **44.33%**.

---

### **(iii) Probability that \( \bar{x} \) is within 3 minutes of the population mean**

Here, we calculate \( P(28.5 \leq \bar{x} \leq 34.5) \):  
\[
P(28.5 \leq \bar{x} \leq 34.5).
\]

For \( \bar{x} = 28.5 \):  
\[
z = \frac{28.5 - 31.5}{1.697} = \frac{-3}{1.697} \approx -1.768.
\]

For \( \bar{x} = 34.5 \):  
\[
z = \frac{34.5 - 31.5}{1.697} = \frac{3}{1.697} \approx 1.768.
\]

The probability is:  
\[
P(28.5 \leq \bar{x} \leq 34.5) = P(-1.768 \leq z \leq 1.768).
\]

Using standard normal distribution tables or a calculator, we find:  
\[
P(-1.768 \leq z \leq 1.768) \approx 0.9228.
\]

Thus, the probability is approximately **92.28%**.

---

### **Summary of Results**
1. The sampling distribution of \( \bar{x} \) is \( N(31.5, 1.697) \).
2. The probability that \( \bar{x} \) is within 1 minute of the population mean is **44.33%**.
3. The probability that \( \bar{x} \) is within 3 minutes of the population mean is **92.28%**.

---

Do you want a detailed breakdown of how to use the \( z \)-table or a plot of the normal distribution?

### Relative Questions
1. How is the standard error of the mean derived, and why does it depend on the sample size?
2. Why does the probability increase when the range around the mean is widened?
3. What happens to the sampling distribution if the sample size increases to 100?
4. How would the probabilities change if the population standard deviation were larger, say 15 minutes?
5. Why is the normal distribution used for the sampling distribution of the mean?

**Tip:** Always confirm whether the sample size is large enough to approximate the sampling distribution as normal.

The mean travel time to work for individuals in a particular city is 31.5 minutes. The population standard deviation is 12 minutes. A sample of 50 residents of the city is selected. (i) Show the sampling distribution of x where x is the sample mean travel time to work for the 50 residents. (ii) What is the probability that the sample mean will be within 1 minute of the population mean? (iii) What is the probability that the sample mean will be within 3 minutes of the population mean? Calculate properly and explain also.

Explore the sampling distribution of the mean travel time to work in a city where the population mean is 31.5 minutes and standard deviation is 12 minutes. Learn how to calculate probabilities that the sample mean is within 1 and 3 minutes of the population mean using the standard normal distribution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation