Question

An automotive company would like to install a new breaking system in a one of its least popular vehicles. The new breaking system promises to be more effective by stopping sooner. To test this claim, the company gathers a sample of 50 cars with the old breaking system and find that the mean stopping distance is 65 feet with a standard deviation of 4 feet. A different sample of the same car model was tested with the new breaking system. This sample of 60 observations yielded a sample mean of 63.5 feet with a standard deviation of 3.5 feet.

Set up null and alternative hypothesis.

Test at the .05 level. What is your conclusion?

For this situation, identify the Type I and Type II errors and the potential issues associated with making each type of error. From a business perspective, which error is worse? What about from the consumer perspective?

Answer 1

Let be the population mean stopping distance for cars with the old braking system.

Let be the population mean stopping distance for cars with the new braking system.

Given:

For: = 65, s₁ = 4, n₁ = 50

For: = 63.5, s₂ = 3.5, n₂ = 60

Since s₁/s₂ = 4/3.5 = 1.14 (it lies between 0.5 and 2) we used the pooled variance.

Since we use the pooled variance, the degrees of freedom = n1 + n2 - 2 = 50 + 60 - 2 = 108

The Hypothesis:

H0: = : The mean braking distance of cars with the old braking system is equal to the mean braking distance of cars with the new braking system.

Ha: > : The mean braking distance of cars with the old braking system is greater than the mean braking distance of cars with the new braking system.

This is a Right tailed test.

The Test Statistic:

The p Value: The p value (Right Tail) for t = 2.10,df = 108,is; p value = 0.0190

The Critical Value:   The critical value (Right tail) at = 0.05, df = 108,tcritical = +1.659

The Decision Rule: If tobserved is > tcritical, Then Reject H0.

Also If the P value is < , Then Reject H0

The Decision:    Since t observed (2.10) is > tcritical (1.659), We Reject H0.

Also since P value (0.0190) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the mean braking distance of cars with the old braking system is greater than the mean braking distance of cars with the new braking system. This means that the new braking system is more effective.

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A Type I error is the incorrect rejection of a true null hypothesis. i.e Actually the mean braking distance is the same for both the braking systems, but we reject this and conclude that the new braking system is better. As we will see, this is not good for the consumer.

Potential issues are

(a) A bad brand name when the product is found to be not as effective.

(b) Consumer might end up spending more for a product which is effectively the same as the old one.

(c) Safety issues in terms of braking distance advertised by the new braking system.

A Type II error occurs when we fail to reject a false null hypothesis. i.e Actually the mean braking distance is for the new braking system is more effective than the old one, but we reject this and conclude that the new braking system is the same as the old braking system. As we will see, this is not good for the business.

Potential issues are:

(a) More money spent on researching something that is already effective.

(b) For the company, The least popular vehicle continues to be the least popular as nothing new has come to change its popularity.

From a Business perspective, The Type II error is worse.

From a consumer point of View, the Type I error is worse.