The amount of water in a bottle is approximately normally distributed with a mean of 2.80 liters with a standard deviation of 0.025 liter. Complete parts​ (a) through​ (c) below. **a.** What is the probability that an individual bottle contains less than 2.78 ​liters?

enter your response here

​(Round to three decimal places as​ needed.)

**b.** If a sample of 4 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.78 ​liters?

enter your response here

​(Round to three decimal places as​ needed.)

**c.** If a sample of 25 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.78 ​liters?

enter your response here

​(Round to three decimal places as​ needed.)


To solve these questions, we can use properties of the normal distribution. Let’s break it down step by step.

We are given:
- Mean \( \mu = 2.80 \) liters
- Standard deviation \( \sigma = 0.025 \) liters

We'll use the standard normal distribution \( Z \), where:

\[
Z = \frac{X - \mu}{\sigma}
\]

where \( X \) is the random variable we are interested in. For part (b) and (c), we use the concept of the **sampling distribution of the sample mean**, which has a standard error \( \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \).

---

### **a.** What is the probability that an individual bottle contains less than 2.78 liters?

We calculate the z-score using the formula:

\[
Z = \frac{2.78 - 2.80}{0.025} = \frac{-0.02}{0.025} = -0.8
\]

Now, using the standard normal table or a calculator:

\[
P(Z < -0.8) = 0.2119
\]

Thus, the probability that an individual bottle contains less than 2.78 liters is **0.212** (rounded to three decimal places).

---

### **b.** If a sample of 4 bottles is selected, what is the probability that the sample mean amount contained is less than 2.78 liters?

For a sample of \( n = 4 \), the standard error is:

\[
\sigma_{\bar{X}} = \frac{0.025}{\sqrt{4}} = \frac{0.025}{2} = 0.0125
\]

Now, calculate the z-score for the sample mean:

\[
Z = \frac{2.78 - 2.80}{0.0125} = \frac{-0.02}{0.0125} = -1.6
\]

Using the standard normal table or a calculator:

\[
P(Z < -1.6) = 0.0548
\]

Thus, the probability that the sample mean is less than 2.78 liters for 4 bottles is **0.055** (rounded to three decimal places).

---

### **c.** If a sample of 25 bottles is selected, what is the probability that the sample mean amount contained is less than 2.78 liters?

For a sample of \( n = 25 \), the standard error is:

\[
\sigma_{\bar{X}} = \frac{0.025}{\sqrt{25}} = \frac{0.025}{5} = 0.005
\]

Now, calculate the z-score for the sample mean:

\[
Z = \frac{2.78 - 2.80}{0.005} = \frac{-0.02}{0.005} = -4.0
\]

Using the standard normal table or a calculator:

\[
P(Z < -4.0) \approx 0.00003
\]

Thus, the probability that the sample mean is less than 2.78 liters for 25 bottles is approximately **0.000** (rounded to three decimal places).

---

### Summary of Answers:
- **a.** Probability for one bottle \( P(X < 2.78) = 0.212 \)
- **b.** Probability for a sample of 4 bottles \( P(\bar{X} < 2.78) = 0.055 \)
- **c.** Probability for a sample of 25 bottles \( P(\bar{X} < 2.78) \approx 0.000 \)

---

Would you like further clarification on any part of this process?

### 5 Related Questions:
1. How is the standard error of the mean calculated for different sample sizes?
2. What is the difference between the probability for a single observation and for a sample mean?
3. How would the probabilities change if the standard deviation were larger?
4. What would happen to the probability if the mean were closer to 2.78 liters?
5. How can we interpret the result of the probability being near 0 in part c?

### Tip:
When dealing with sample means, always remember that increasing the sample size reduces the standard error, making it less likely for the sample mean to deviate significantly from the population mean.

The amount of water in a bottle is approximately normally distributed with a mean of 2.80 liters with a standard deviation of 0.025 liter. Complete parts​ (a) through​ (c) below. a. What is the probability that an individual bottle contains less than 2.78 ​liters? b. If a sample of 4 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.78 ​liters? c. If a sample of 25 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.78 ​liters?

Learn how to calculate the probability that the water content in a bottle is less than 2.78 liters using the normal distribution. Explore how the sample size affects the probability and how the central limit theorem applies to the sample mean. This problem involves a mean of 2.80 liters and standard deviation of 0.025 liters, and is suitable for college-level statistics.

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