https://r2.erweima.ai/mathbot/image/17258153157184uryotu4.jpg

Let's solve the problem presented in the image step by step.

### Given Data:
- **Control group** mileages: 13.8, 19.9, 15.1, 18.6, 15.5, 14.6, 17.5, 13.2, 19.7, 12.3
- **With additive** mileages: 15.9, 20.4, 14.0, 18.8, 17.7, 17.5, 17.9, 13.1, 19.3, 22.2

#### Part (i): Calculate a 95% confidence interval for the difference between the mean mileages.
We are tasked with finding a 95% confidence interval for the difference in means between the control group and the additive group.

To do this, we'll calculate:
- The mean and standard deviation of both groups.
- The standard error of the difference.
- The t-value for a 95% confidence level.
- Finally, we'll use this information to calculate the confidence interval.

I'll calculate these values now.### Part (i): 95% Confidence Interval for the Difference in Mean Mileages
- **Mean (Control group)**: 16.02
- **Mean (With additive)**: 17.68
- **Difference in means**: \( 1.66 \)

The 95% confidence interval for the difference between the means is:
\[
\text{CI} = ( -0.93, 4.25 )
\]
This interval suggests that the additive may not necessarily improve mileage since the confidence interval contains zero, meaning there is no significant difference between the groups.

#### Assumptions:
- Both samples are independent.
- The mileages follow a normal distribution.
- Equal variances between the two groups (pooled standard deviation).

#### Comment:
Since the confidence interval includes 0, we cannot conclude with 95% confidence that the additive improves mileage.

---

#### Part (ii): Calculate the minimum sample size so that the width of a 95% confidence interval for the difference in means is less than 4, assuming the samples have the same sizes and equal variances.

We'll solve this next by ensuring the width of the confidence interval (twice the margin of error) is less than 4.### Part (ii): Minimum Sample Size Calculation
The calculation suggests that a sample size of less than 1 (approximately 0.84) is required to achieve a confidence interval width of less than 4. This indicates that, given the current data's variability, the sample size is already large enough to satisfy the condition. However, in practice, sample sizes should be whole numbers, so at least **1 car** per group is sufficient.

---

#### Part (iii): Can we apply a paired t-test in part (i)? Justify.

A **paired t-test** compares two related groups. Here, the control group and additive group consist of different cars, so they are independent samples. Therefore, a **paired t-test is not appropriate**. Instead, the independent two-sample t-test, which we used, is suitable for this scenario.

---

Would you like further details or have any questions? Here are some related questions you might want to explore:

1. How would the result change if variances between the groups were unequal?
2. How is the t-distribution used for calculating confidence intervals?
3. What is the impact of increasing the sample size on the confidence interval width?
4. How does the normality assumption affect the validity of the t-test?
5. What are the conditions under which a paired t-test would be appropriate?

**Tip:** When comparing two groups, always verify whether the samples are independent or related, as this determines the correct statistical test.

An automobile company plans to introduce a fuel additive to increase the mileage in cars. A random sample of 20 cars was used to test the mileage with and without the additive. Data was collected, and the following tasks are required: (i) Calculate a 95% confidence interval for the difference between the mean mileages. (ii) Calculate the minimum sample size required to achieve a 95% confidence interval with a width less than 4. (iii) Determine if a paired t-test can be applied and justify.

Calculate the 95% confidence interval for the difference in mileages between cars with and without fuel additives, and determine the minimum sample size for a 95% CI width of less than 4. Also, assess if a paired t-test is appropriate for this scenario.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation