Question

An organization surveyed 617 high school seniors from a certain country and found that 327 believed they would not have enough money to live comfortably in college. The folks at the organization want to know if this represents sufficient evidence to conclude a majority​ (more than 50​%) of high school seniors in the country believe they will not have enough money in college.

​(a) If the researcher decides to test this hypothesis at the alpha (α) equals=0.05 level of​ significance, compute the probability of making a Type II​ error, beta β​, if the true population proportion is 0.56. What is the power of the​ test?

​(b) Redo part​ (a) if the true population proportion is 0.59.

Answer 1

We are asked to compute the probability of making a type II error and the power of the test given the following values.

n=617, x=327=>p=x/n=327/617=0.53

P=50%=0.50.

First we formulate the null and the alternate hypotheses to gain an understanding what type II error means,

Null hypothesis

H₀:P=0.50

i.e., fifty percent of high school seniors in the country believe that they will not have enough money in college.

Alternate hypothesis

H₁:P>0.50

i.e., majority of high school seniors in the country believe that they will not have enough money in college.

So, type II error in our case would be concluding that fifty percent of high school seniors in the country believe that they will not have enough money in college when in fact the null hypothesis is false i.e., failure to accept the hypothesis that majority of high school seniors in the country believe that they will not have enough money in college.

We now calculate type II error probability given that the true population proportion is 0.56 i.e., P=0.56=>Q=1-P=1-0.56=0.44.

Now that we have obtained the z-score, we can also find the probability by looking at the standard normal tables for the row value of -1.50. This value is 0.0668

Hence the probability of type II error() is .0668.

Power of the test is defined as the probability of rejecting null hypothesis when the null hypothesis is false i.e., power of test=1-.

Hence the power of the test is 1-0.0668=0.9332.

Therefore probability of type II error() is 0.0668 and power of the test is 0.9332 when the true population proportion is 0.56.

b) We are asked to compute the type II error and the power of the test given true population is 0.59

The value corresponding to the row value of -3.03 is 0.0012 from the standard normal tables.

Hence the probability of type II error() is 0.0012.

Power of the test in this case is 1-0.0012=0.9988.

Therefore probability of type II error when the true population is 0.59 is 0.0012 and power of the test is 0.9988