Question

A major automobile company claims that its New electric powered car has an average range of more that 100 miles. A random sample of 50 new electric cars was selected to test the claim. Assume that the population standard deviation is 12 miles. A 5% level of significance will be used for the test.

A) What would be the consequences of making a Type II error in this problem?

B) Compute the Probability of making a Type II error if the true population mean is 105 miles.

C) What is the maximum probability of making a Type I error in this problem?

Please Note: A hypothesis test answer must contain: a Null and an Alternate Hypothesis, a computed value of the test statistic, a critical value of the test statistic, a Decision , and a Conclusion.

Answer 1

a) The type II error happens when we retain a false null hypothesis. Here the null hypothesis is that the average range is not more than 100 miles i.e it is 100 miles. Therefore the consequence of a type II error would be that we would conclude that the average range is 100 miles although in reality it is more than 100 mile.

b) Given the true population mean as 105 miles, the distribution of the sample mean by Central Limit theorem is given as:

Now for 5% level of significance, we are rejecting the null hypothesis if the sample mean is more than x where x is obtained as such that:
P(X > x) = 0.05

From standard normal tables, we have:
P(Z > 1.645) = 0.05

Therefore, the value of x here is computed as:

Now the probability of type II error here is computed as:

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.0966 is the required probability of type II error here.

c) The maximum probability of type I error is given as the level of significance for the test which is given to 0.05 here. Therefore 0.05 is the required probability here.