91Ó°ÊÓ

Gamma Corporation is considering the installation of govemors on cars driven by its sales staff. These devices would limit the car speeds to a preset level, which is expected to improve fuel economy. The company is planning to test several cars for fuel consumption without governors for 1 week. Then governors would be installed in the same cars, and fuel consumption will be monitored for another week. Gamma Corporation wants to estimate the mean difference in fuel consumption with a margin of error of estimate of 2 mpg with a 90 \% confidence level. Assume that the differences in fuel consumption are normally distributed and that previous studies suggest that an estimate of \(s_{d}=3 \mathrm{mpg}\) is reasonable. How many cars should be tested? (Note that the critical value of \(t\) will depend on \(n\), so it will be necessary to use trial and error.)

Step by step solution

01

Define the variables

First, define the variables from the given problem. Here, \(d = 2 \, mpg\) is the margin of error, \(s_{d}=3 \, mpg\) is the estimated standard deviation, and the confidence level is \(90\% = 1 - \alpha = 0.9\), so \(\alpha = 0.1\). Since this is a two-tailed test, \(\alpha/2 = 0.05, \, \text{and then}\, t_{\alpha/2} = t_{0.05}\). The value of \(t_{0.05}\) will depend on the degrees of freedom which is \(df = n - 1\). Since \(n\) is unknown, the exact value of \(t_{0.05}\) can't be determined at this stage.

02

Setup the equation

We'll use the formula for the sample size, which for this problem is \(n = (\frac{t_{\alpha/2} \times s_{d}}{d})^2\). Substituting the known values we have \(n = (\frac{t_{0.05} \times 3}{2})^2\). We can't solve for \(n\) yet because \(t_{0.05}\) depends on \(n\).

03

Iterations to find \(n\)

As the exact \(n\) number is unknown, we have to make guesses for \(n\) and, as a result, refine the \(t_{0.05}\) values. Start by guessing \(n=10\), with \(df=9\), check the t-table for \(t_{0.05,9}=1.833\). Substitute this value in the formula and find the calculation of \(n\). Keep adjusting the guess for \(n\) until the computed \(n\) and the guessed \(n\) are close enough.

04

Final calculation

By going through the loop of steps and refining the guesses, we can find the minimum sample size required to ensure a 90 \% confidence level and a margin of error of 2 mpg.

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Key Concepts

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

Margin of Error

The margin of error is a crucial concept when estimating statistics. It represents the range within which the true value of a parameter (like the mean) is expected to lie. In our exercise, the margin of error is 2 miles per gallon (mpg). This means that the estimated mean difference in fuel consumption might vary by up to 2 mpg from the true mean difference.

The margin of error is determined by the standard deviation of the data, the sample size, and the confidence level. In simpler terms, it's how much wiggle room we allow around our estimated mean. A smaller margin of error means our estimate is more precise, requiring a larger sample size or greater certainty about other factors.

• Wide margins mean less precision.
• Narrow margins require more data or are more precise.

Confidence Level

Confidence level tells us how sure we are that the true parameter falls within our margin of error. In this exercise, we are using a 90% confidence level. This means there's a 90% chance that the true mean difference in fuel consumption is within our calculated range, given our sample data.

Higher confidence levels lead to broader intervals, as we need more certainty, which can require a larger sample size.

• A 90% confidence level is commonly used for moderate assurance.
• Higher levels, like 95%, offer more certainty but also widen the estimation range, possibly needing larger samples.

Selecting the right confidence level depends on how much certainty is needed for the conclusions being drawn.

T-Distribution

The t-distribution is a statistical distribution used when the sample size is small and the population variance is unknown. It helps estimate the population parameters, especially when dealing with smaller sample sizes. In this exercise, the t-distribution is key to estimating the sample size needed to achieve our margin of error and confidence level.

The shape of the t-distribution changes based on degrees of freedom, which is related to sample size ( - 1 in this case). Unlike the standard normal distribution, the t-distribution is broader, accommodating the greater variability in estimates from smaller samples.

• T-distribution adjusts for small sample sizes.
• Critical value of t changes with degrees of freedom.

It's essential for accurate sample size calculations in our exercise.

Fuel Efficiency Testing

Fuel efficiency testing, like in this exercise, involves assessing how modifications to vehicles impact fuel consumption. Gamma Corporation is evaluating whether installing governors can improve this.

The process includes testing fuel consumption before and after the modifications, measuring any differences. By estimating the mean difference, the company can infer the effectiveness of the governors. This testing relies on carefully controlled conditions to ensure accuracy and relevance.

• Comparative testing between normal and modified states.
• Statistical analysis determines the modification's impact.

Understanding the exact impact on fuel efficiency helps in making data-driven decisions for vehicle usage strategies.