91Ó°ÊÓ

Gamma Corporation is considering the installation of governors 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

Understand the given parameters

From the problem, identify the standard deviation \(s_d = 3\) mpg, margin of error E = 2 mpg, and confidence level which is 0.90. Even though the problem does not provide a significance level \(\alpha\), knowing that confidence level = 1 - \(\alpha\), \(\alpha\) can be found as \(\alpha = 1 - 0.90 = 0.10\). Since this is a two-tailed test, \(\alpha/2 = 0.10/2 = 0.05\). The degrees of freedom are unknown since they depend on the sample size \(n\).

02

Use the margin of error formula

The formula for the margin of error in this case, using a t distribution, is: E = \(t_{\alpha/2}\) * \(\frac{s_d}{\sqrt{n}}\), where \(t_{\alpha/2}\) is the critical t-value for our chosen significance level. Solving this formula for \(n\) gives: \(n = (\frac{t_{\alpha/2} * s_d}{E})^2\)

03

Use the trial and error method

The difficult part here is that the critical t-value depends on the sample size, but we're trying to find the sample size. The best way to handle this is to start with a guess of the degrees of freedom, compute the sample size, and then update the guess of the degrees of freedom until it's close to \(n - 1\). We start by assuming the degrees of freedom to be 30 (a common assumption for large samples), which gives a \(t_{\alpha/2}\) = 2.042 (This can be found using t-table). Substituting \(t_{\alpha/2} = 2.042\), \(s_d = 3\) and \(E = 2\) into the formula gives a sample size of \(n = (2.042 * 3/2)^2 = 10.59\). However, we cannot have a fractional sample size so we will round off to the nearest whole number. Rounding 10.59 to nearest whole number gives us 11. However, degrees of freedom should be \(n-1 = 11-1 = 10\). So our guess was close enough.

04

Validate the solution

Since our new degrees of freedom is 10, we need to get the new critical value of \(t_{\alpha/2}\) for the degrees of freedom 10 and then recalculate the sample size. This gives \(t_{\alpha/2}\) = 2.23, hence recalculating our \(n\) value: \(n = (2.23 * 3/2)^2 = 14.06\). Rounding off to the nearest whole number gives us 14. Our new degrees of freedom = 14 - 1 = 13. The critical value for 13 degrees of freedom is \(t_{\alpha/2}\) = 2.160. We recalculated \(n\) again: \(n = (2.160 * 3/2)^2 = 13.53\) hence our sample size is 14. We could keep refining this, but our original guess of 11 was close enough to give us the correct answer generally. The answer implies that 14 cars should be tested in order to obtain a margin of error of 2 mpg at a confidence level of 90%.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values that is used to estimate a population parameter, such as the mean difference in fuel consumption for Gamma Corporation's cars with and without governors. This interval is derived from the statistics of a sample, and it provides a measure of certainty around the sample mean. The confidence level, which in this case is 90%, tells us how sure we can be that the interval incorporates the true mean difference.

• The width of a confidence interval is affected by the sample size and variability of the data: larger sample sizes or less variability result in narrower intervals.
• In practical terms, a 90% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each, we would expect approximately 90 of those intervals to contain the true mean difference.

Understanding the concept of a confidence interval is crucial because it guides us in interpreting results and making decisions based on sample data.

Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. In the context of Gamma Corporation's project, the margin of error refers to the desired precision of the estimate of the mean difference in mpg when governors are installed. Here, it is specified as 2 mpg.

• The margin of error decreases as the sample size increases, assuming other factors remain constant. This relationship is important because it helps determine how many cars need to be tested to achieve a reliable estimate.
• The formula used to compute the margin of error involves the critical value from the t-distribution and the standard deviation of the differences. The smaller the margin of error, the more accurate the results are deemed to be.

Having a clear understanding of the margin of error allows for improved planning and decision-making in experimental designs.

t-Distribution

The t-distribution is a probability distribution that is used when estimating population parameters, particularly when dealing with small sample sizes or unknown population standard deviations. In this exercise, the Gamma Corporation uses the t-distribution to find the critical t-value.

• The t-distribution is similar to the normal distribution but with heavier tails, which means it is more prone to producing values that fall far from its mean.
• This property makes it suitable for small sample sizes or when the population standard deviation is not known, as is the case in the fuel consumption tests.

Understanding the t-distribution is important because it provides the basis for estimating the sample size and critical values that influence the margin of error and confidence intervals.

Mean Difference Estimation

Mean difference estimation is central to evaluate the impact of installing governors on cars in this exercise. The goal is to calculate an estimated mean difference in fuel consumption before and after governors are installed. This estimation helps infer if the governors significantly impact fuel consumption on average.

• The mean difference estimation relies on the assumption that the differences are normally distributed as stated in the exercise. This assumption is crucial as it justifies using the t-distribution for deriving estimates.
• Accurate mean difference estimation allows corporations like Gamma to make informed decisions about potential costs and benefits. By analyzing the mean difference, the company can confidently assess whether installing the governors leads to improved fuel efficiency and justifies the implementation across its fleet.

Therefore, mean difference estimation provides actionable insights into operational changes.