A car manufacturer is interested in conducting a study to estimate the mean stopping distance for a new type of brakes when used in a car that is traveling at 60 miles per hour. These new brakes will be installed on cars of the same model and the stopping distance will be observed. The cost of each observation is $100. A budget of $12,000 is available to conduct the study and the goal is to carry it out in the most economical way possible. Preliminary studies indicate that s=12 feet for stopping distances.
(a) Are sufficient funds available to estimate the mean stopping distance to within 2 feet of the true mean stopping distance with 95% confidence? Explain your answer.
(b) A regulatory agency requires a 95% level of confidence for an estimate of mean stopping distance that is within 2 feet of the true mean stopping distance. The car manufacturer cannot exceed the budget of $12,000 for the study. Discuss the consequences of these constraints.
a) there are not sufficient funds. THe required sample size is 139 cars, and if each car will cost $100, then the total cost for estimating the mean stopping distance to within 2 feet of the true mean is $139,000, which is more than the budget will allow.
b) $12,000 is not enough money to be within 2 feet of the true mean with 95% confidence. $12,000 only allows for 120 cars to be tested, and 139 are required. Given these constraints, the consequences are that the company will not be 95% confident on the mean stopping distance for their brakes, which could lead to car accidents and possible lawsuits.
To estimate the mean stopping distance for the new type of brakes with 95% confidence and an accuracy of within 2 feet, we need to determine if there is enough budget available to conduct the study efficiently. Let's walk through the calculations for both parts (a) and (b) of the question:
(a) To estimate the mean stopping distance to within 2 feet of the true mean stopping distance with 95% confidence, we can use the formula for the sample size required in a mean estimation problem:
n = (Z * s / E)^2
Where:
n = sample size needed
Z = Z-value for the desired level of confidence (95% confidence corresponds to a Z-value of approximately 1.96)
s = standard deviation of the stopping distances (given as s = 12 feet)
E = desired margin of error (2 feet)
Plugging in the values:
n = (1.96 * 12 / 2)^2
n = 89.86
Since a sample size needs to be a whole number, we round up to the nearest whole number:
n = 90
So, ideally, we would need a sample size of 90 observations to estimate the mean stopping distance to within 2 feet with 95% confidence.
(b) To determine if the available budget of $12,000 is sufficient for the study, we need to consider the cost of each observation. Given that each observation costs $100, we can calculate the maximum number of observations we can afford:
maximum number of observations = available budget / cost per observation
maximum number of observations = $12,000 / $100
maximum number of observations = 120
Since the maximum number of observations we can afford is 120, and we ideally need a sample size of 90, it is possible to conduct the study within the given budget.
However, there are consequences to consider due to these constraints. With a smaller sample size (90) than the maximum possible (120), there may be reduced precision in the estimation. A larger sample size usually leads to a more accurate estimate of the population mean. By restricting the budget, there is a trade-off between cost and precision. The resulting estimate might not be as reliable as it would have been with a larger sample size. The manufacturer needs to carefully consider these trade-offs and the potential implications for their decision-making process.
To answer these questions, we need to calculate the sample size required to estimate the mean stopping distance within a certain margin of error and confidence level. We can use the formula for the sample size needed for estimating a population mean:
n = (Z * σ / E)^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired confidence level
- σ is the population standard deviation
- E is the desired margin of error
(a) To estimate the mean stopping distance within 2 feet of the true mean stopping distance with 95% confidence, we need to calculate the required sample size.
Given:
- Confidence level (1 - α) = 0.95
- Margin of error (E) = 2 feet
- Population standard deviation (σ) = 12 feet
Since no information is given about the population size, we can assume it is large enough that we can use the normal distribution instead of the t-distribution. The z-score corresponding to a 95% confidence level is approximately 1.96.
Using the formula, we can calculate the required sample size:
n = (1.96 * 12 / 2)^2
n = (23.52 / 2)^2
n = 11.76^2
n ≈ 138.17
The required sample size is approximately 139 observations.
The cost per observation is $100, so the total cost will be:
Total cost = Sample size * Cost per observation
Total cost = 139 * $100
Total cost = $13,900
Since the total cost exceeds the available budget of $12,000, there are not sufficient funds available to estimate the mean stopping distance within 2 feet of the true mean stopping distance with 95% confidence.
(b) With a budget constraint of $12,000 and the requirement to have a 95% confidence level and a margin of error within 2 feet, we need to find a solution that fits within the given budget.
Since the cost per observation is fixed at $100, we need to reduce the sample size to fit within the budget. However, reducing the sample size will increase the margin of error, affecting the precision of the estimate.
To find a feasible solution, we can try reducing the sample size until the total cost is within the budget while still maintaining a reasonable margin of error.
Let's test a few sample sizes and calculate the corresponding total cost and margin of error:
For n = 100:
Total cost = 100 * $100 = $10,000
Margin of error = (1.96 * 12) / √100 ≈ 2.35 feet
For n = 80:
Total cost = 80 * $100 = $8,000
Margin of error = (1.96 * 12) / √80 ≈ 2.63 feet
For n = 60:
Total cost = 60 * $100 = $6,000
Margin of error = (1.96 * 12) / √60 ≈ 3.06 feet
As we can see, reducing the sample size to fit within the budget leads to a larger margin of error, which reduces the precision of the estimate. Therefore, the consequence of these constraints is that the car manufacturer will have to accept a larger margin of error in estimating the mean stopping distance.