A botanist creates a test of hypothesis to decide if the stem diameters for a hybrid plant are larger than those for standard plants.Standard plants have a mean stem diameter of 39 mm and a standard deviation of 4 mm.The botanist randomly samples 30 hybrid plants and records the stem diameters for each.What is the probability of a type II error if the population mean diameter of the hybrid plants is 41 mm and she assumes the standard deviation is still 4 mm? (Use α = 0.05,)
A) 0.9500
B) 0.9999
C) 0.1370
D) 0.2181

Question

Quizplus · Accepted Answer

The probability of a type II error is the probability of failing to reject the null hypothesis when it is false. In this case, the null hypothesis is that the mean stem diameter of the hybrid plants is 39 mm, and the alternative hypothesis is that the mean stem diameter of the hybrid plants is greater than 39 mm.The probability of a type II error is given by the area under the normal distribution curve to the left of the critical value, which is the value that separates the rejection region from the non-rejection region. The critical value is determined by the significance level, which is the probability of rejecting the null hypothesis when it is true. In this case, the significance level is 0.05.The critical value for a one-tailed test with a significance level of 0.05 is 1.645. The standard deviation of the population is 4 mm, and the sample size is 30. The mean of the sample is 41 mm.The z-score for the sample mean is (41 - 39) / (4 / sqrt(30)) = 2.121.The probability of a type II error is the area under the normal distribution curve to the left of the z-score of 2.121, which is 0.1370.

A Botanist Creates a Test of Hypothesis to Decide If