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Chapter 9: Problem 18

For a simple random sample of forty 2005 Honda Accords (4 cylinder, 2.4 liter, 5 -speed automatic), the mean gas mileage was 23 miles per gallon with a standard deviation of 1.5 miles per gallon. Construct a \(95 \%\) confidence interval for the mean gas mileage of similar 2005 Honda Accords. Interpret the interval.

Short Answer

Expert verified
The 95% confidence interval is (22.535, 23.465) mpg.

Step by step solution

01

- Understand the Problem

Given a sample size of 40 Honda Accords, with a mean gas mileage of 23 mpg and a standard deviation of 1.5 mpg, the task is to construct a 95% confidence interval for the true mean gas mileage.
02

- Identify the Formula

To calculate the confidence interval, use the formula for the confidence interval for the mean, which is: \[ CI = \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]where \(\bar{x}\) is the sample mean, \(z\) is the z-value for a 95% confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size.
03

- Find the Critical Value

For a 95% confidence interval, the z-value (critical value) corresponding to the 95% confidence level is approximately 1.96.
04

- Plug in the Values

Substitute the given values into the formula:\[ CI = 23 \pm 1.96 \frac{1.5}{\sqrt{40}} \]
05

- Calculate the Standard Error

Calculate the standard error (SE):\[ SE = \frac{1.5}{\sqrt{40}} \approx 0.237 \]
06

- Compute the Margin of Error

Compute the margin of error by multiplying the critical value by the standard error:\[ ME = 1.96 \times 0.237 \approx 0.465 \]
07

- Construct the Confidence Interval

Add and subtract the margin of error from the sample mean to construct the confidence interval:\[ CI = 23 \pm 0.465 \]\[ CI = (22.535, 23.465) \]
08

- Interpret the Confidence Interval

The 95% confidence interval for the mean gas mileage of similar 2005 Honda Accords is between 22.535 and 23.465 miles per gallon. This means we are 95% confident that the true mean gas mileage lies within this interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mean gas mileage
The **mean gas mileage** tells us about the average fuel efficiency of the cars in our sample. In the stated problem, the sample mean is 23 miles per gallon (mpg). This means that on average, these 2005 Honda Accords travel 23 miles for every gallon of gas. The mean is a central value that gives us a good idea of typical performance. However, real-world data varies, making it essential to use this along with other statistical measures.
sample size
In the problem, the **sample size** is 40, referring to the number of 2005 Honda Accords whose gas mileage was recorded. The sample size is crucial because it affects the reliability and accuracy of our confidence interval. A larger sample size generally provides a more precise estimate of the population mean. The sample size is denoted by the letter 'n' and plays a key role in calculating the standard error. Simply put, more data points lead to a narrower confidence interval, indicating a more reliable mean estimate.
standard deviation
The **standard deviation** measures how much the gas mileage values in the sample deviate from the mean, or how spread out the values are. In this example, the standard deviation is 1.5 mpg. This means that the gas mileage of the different Honda Accords varies, on average, by 1.5 miles per gallon from the mean. The standard deviation is denoted by the Greek letter sigma (蟽). It鈥檚 an essential factor in calculating the standard error and subsequently the margin of error for our confidence interval.
95% confidence level
A **95% confidence level** indicates that if we were to draw 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the true population mean. This level of confidence is commonly used in statistical analysis, and its corresponding z-value is approximately 1.96. This z-value is used to calculate the margin of error. A higher confidence level would result in a wider interval, while a lower confidence level would generate a narrower interval.
margin of error
The **margin of error** quantifies the amount of random sampling error in our estimate. It represents the range above and below the sample mean in which the true population mean is likely to fall. In this case, the margin of error is calculated to be approximately 0.465 mpg. This was computed by multiplying the z-value (1.96) by the standard error (0.237). The margin of error helps in constructing the confidence interval, which in this case is from 22.535 mpg to 23.465 mpg, ensuring a 95% confidence that the true mean gas mileage falls within this range.

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