im so confused! please help!
3. 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'll need to perform some calculations.
(a) To estimate the mean stopping distance with a 95% confidence level, we can use the formula:
Sample Size (n) = (Z-score * Standard Deviation / Margin of Error)^2
Given that the standard deviation (s) is 12 feet and the margin of error (E) is 2 feet, and assuming a 95% confidence level, we need to find the appropriate Z-score for a 95% confidence level. The Z-score for a 95% confidence level is approximately 1.96.
Using these values, we can compute the required sample size (n):
n = (1.96 * 12 / 2)^2
n = 42.336
Since the sample size must be a whole number, we'll take the ceiling of this value, which gives n = 43.
The cost per observation is $100, so the total cost of the study can be calculated by multiplying the sample size by the cost per observation:
Total cost = Sample Size * Cost per Observation
Total cost = 43 * $100
Total cost = $4,300
Since the available budget is $12,000, which is greater than the total cost of $4,300, sufficient funds are available to estimate the mean stopping distance to within 2 feet of the true mean stopping distance with 95% confidence.
(b) If the regulatory agency requires a 95% confidence level and a margin of error within 2 feet of the true mean stopping distance, and the budget is limited to $12,000, there will be some consequences.
The cost per observation is $100, and the manufacturer's budget is $12,000. Therefore, the maximum number of observations (sample size) that can be taken is:
Maximum sample size = Budget / Cost per Observation
Maximum sample size = $12,000 / $100
Maximum sample size = 120
However, from the previous calculation, we determined that we needed a sample size of 43 to estimate the mean stopping distance with a margin of error within 2 feet and a 95% confidence level. Thus, it is not possible to meet the regulatory agency's requirement within the given budget constraints.
The consequences of these constraints could include:
1. The study might need to be modified: The manufacturer may have to plan for a smaller sample size or compromise on the desired margin of error or confidence level.
2. Limited precision: With a smaller sample size, the estimate of the mean stopping distance may have a larger margin of error, resulting in less precise results.
3. Potential for errors: With a smaller sample size, there is an increased risk of sampling error, which could affect the accuracy of the estimate.
4. Uncertainty in decision-making: The manufacturer and regulatory agency may need to make important decisions based on an estimate with a larger margin of error, which could lead to uncertainty in determining the effectiveness of the new brakes.
In summary, the budget constraints will limit the sample size and compromise the precision and accuracy of the estimate of the mean stopping distance, potentially affecting decision-making and regulatory compliance.