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.

To determine if there are sufficient funds available to estimate the mean stopping distance to within 2 feet of the true mean stopping distance with 95% confidence, we need to consider the sample size required for a given level of confidence.

(a) To estimate the sample size required, we can use the formula:

n = (Z * (s / E))^2

Where:
- n is the required sample size
- Z is the Z-score corresponding to the desired level of confidence (95% confidence corresponds to a Z-score of 1.96)
- s is the standard deviation of the population (preliminary studies indicate s = 12 feet)
- E is the maximum margin of error (in this case, E = 2 feet)

Plugging in the values:

n = (1.96 * (12 / 2))^2
n = (1.96 * 6)^2
n = 37.49^2
n ≈ 1405

So, we would need a minimum sample size of 1405 observations to estimate the mean stopping distance to within 2 feet of the true mean stopping distance with 95% confidence.

Now, let's check if the available budget of $12,000 is sufficient for this study.

(b) To calculate the cost of the study, we multiply the sample size by the cost per observation:

Cost = n * Cost per observation
Cost = 1405 * $100
Cost = $140,500

The estimated cost ($140,500) exceeds the available budget ($12,000), so it is not possible to conduct the study with the given budget while meeting the desired level of confidence and margin of error requirements.

Consequences of these constraints could include the need to reduce the sample size, which would result in a larger margin of error or lower confidence level. Alternatively, the budget could be increased to meet the desired requirements. The car manufacturer would need to weigh the trade-offs between cost, precision, and confidence when making their decision.