Question

1. NASA has been working with utility companies to set up expensive power generating windmills. In order for a site to be effective, the average wind speed must be more than 20 mph. At a specific site, NASA took a sample of forty-five wind speed readings. The sample average wind speed is 21.7 mph with a standard deviation of 4.2 mph. NASA will only build the windmill if they are sure that the true average wind speed is greater than 20 mph.

1a. Should they build the windmill? Justify by conducting a hypothesis test at α = 0.05.

1b. Describe in the words of the problem what making a Type I error would be and its likely consequences.

1c. Describe in the words of the problem what making a Type II error would be and its likely consequences.

Answer 1

Solution:-

Given

n = 45

1(a)

The variable here is wind speed, i.e., x is the wind speed

= 21.7 mph

S = 4.2 mph

= 20 mph

we have to test

vs

The test statistic to test the above hypothesis is one-sample t-test.

with n - 1 of f.

t = 2.72 with 44 df

The p-value for above test is 0.0047.

The obtained p-value is < 0.05 which suggests that we have strong evidence against H0 yo reject it and we can conclude that sample average wind speed is more than 20 mph and also we can consult that they should built the windmill.

1(b):

The type-I error is defined as rejecting H0 when it is true.

In this problem if actual windspeed is not greater than 20 mph and still we reject the null hypothesis on the basis of sample evidences then we are making a type- I error and it will affect the power generation of windmills. By committing type-I error we are wrongly setting up the power generating windmills at the site and if the windmills are not greater than 20 mph then the windmills will not be effective.

1(c):

The type-II error is defined as accepting H0 when it is fase.

In this problem, if the actual windspeed is greater than 20 mph and we accept the null hypothesis that the windspeed is not greater than 20 mph on the basis of sample evidences then we are making a type-II error. By committing a type-II error, we will not suggest to build a power generating windmill on the site while it is suitable to build a power generating windmill on the site.