You would like to construct a 95% confidence interval to estimate the population mean time it takes drivers to react following the application of brakes by the driver in front of them. You take a random sample of reaction time measurements and compute their mean to be 1.6 seconds and their standard deviation to be 0.5 seconds.

(a) What is the best point estimate, based on the sample, to use for the population mean?

seconds

X

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(b) For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 95% confidence interval for the population mean.

[In the table, Z refers to a standard normal distribution, and refers to at distribution.)

Sampling scenario

The sample has size 85, and it is from a non-normally distributed population.

The sample has size 18, and it is from a normally distributed population with a known standard deviation of 0.45.

The sample has size 11, and it is from a normally distributed population with an unknown standard deviation.

Z

Could use either Zor!

Unclear

X

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### (A) What is the best point estimate, based on the sample, to use for the population mean?

The best point estimate for the population mean is the sample mean. Given that the sample mean is 1.6 seconds, this is the best estimate.

- **Answer:** \( \text{1.6 seconds} \)

### (B) Determine which distribution should be used to calculate the critical value for the 95% confidence interval for the population mean.

#### 1. **Sampling Scenario 1:**
   - **Sample size:** 85
   - **Population distribution:** Non-normally distributed

   **Distribution to Use:** Since the sample size is large (typically \( n > 30 \)), the Central Limit Theorem applies, allowing us to use the **z-distribution** for the critical value, even though the population is non-normally distributed.

   - **Answer:** \( Z \)

#### 2. **Sampling Scenario 2:**
   - **Sample size:** 18
   - **Population distribution:** Normally distributed with a known standard deviation of 0.45

   **Distribution to Use:** When the population is normally distributed and the population standard deviation is known, the **z-distribution** should be used, regardless of the sample size.

   - **Answer:** \( Z \)

#### 3. **Sampling Scenario 3:**
   - **Sample size:** 11
   - **Population distribution:** Normally distributed with an unknown standard deviation

   **Distribution to Use:** When the sample size is small (\( n \leq 30 \)), the population is normally distributed, and the population standard deviation is unknown, the **t-distribution** should be used.

   - **Answer:** \( t \)

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Would you like more details or have any questions?

Here are some related questions:
1. How would the confidence interval change if the sample mean was 2.0 seconds instead of 1.6 seconds?
2. What is the formula to calculate the confidence interval for the population mean?
3. How does the sample size affect the width of the confidence interval?
4. What are the implications of using a t-distribution instead of a z-distribution?
5. Can you explain why the Central Limit Theorem allows us to use the z-distribution for large samples, even if the population distribution is not normal?

**Tip:** Always check the sample size and whether the population standard deviation is known to determine whether to use the z-distribution or t-distribution when constructing confidence intervals.

Learn how to construct a 95% confidence interval for estimating the population mean reaction time following the application of brakes by the driver in front. Includes detailed explanations for different sampling scenarios and the use of z-distribution and t-distribution.

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