Question

1)The life in hours of a battery is known to be approximately normally distributed, with standard deviation 1.25 hours. A random sample of 10 batteries has a mean life of 40.5 hours. A. Is there evidence to support the claim that battery life exceeds 40 hours ? Use α= 0.05.

B. What is the P-value for the test in part A?

C. What is the β-error for the test in part A if the true mean life is 42 hours?

D. What sample size would be required to ensure that doesnot exceed 0.10 if the true mean life is 44 hours?

2) To study the elastic properties of a material manufactured by two methods, the deflection in mm of 2x15 elastic bars made from the two materials are measured, and the are shown in the table:

Method 1	Method 2
206	177
188	197
205	206
187	201
194	180
193	176
207	185
185	200
189	197
213	192
192	198
210	188
194	189
178	203
205	192

A) Construct Relative Frequency Plots for the two sets of data, use 6 bins of increment of 10 starting from 170mm, and then create a Cumulative Distributions of the two data sets. By comparing the slopes of the cumulative distributions, do you think these plots provide support of the assumptions of normality and equal variances for the populations of the two sets?

B) Assuming equal variances of population, do the data support the claim that the mean deflection from method 2 exceeds that from method 1? Use α= 0.05

C) Resolve part B assuming that the variances of populations are not equal.

D)Suppose that if the mean deflection from method 2 exceeds that of method 1 by as much as 5mm, it is important to detect this difference with probability at least 0.90. Is the choice of n1=n2=15 in part A of this problem adequate

E) Make a test that can be used to tell which data set is more accurate than the other (whether method 1 is more accurate or from method two) .

Answer 1

Dear student we can provide you with a solution of one type of question at a time.

a) Since the population standard deviation is given we will use Z test for testing mean.

Hence We have no evidence to support the claim that battery life exceeds 40 hours.

b) This is a right-tailed test P-value is

c) This is a right-tailed test.

We will fail to reject the null (commit a Type II error) if we get a Z statistic of less than 1.64.

This 1.64 Z-critical value corresponds to some X critical value ( X critical), such that

So I will incorrectly fail to reject the null as long as a draw a sample mean that is less than 40.65. To complete the problem what I now need to do is compute the probability of drawing a sample mean less than 40.65.

given µ = 42 . Thus, the probability of Type II () error is given by

d)

probability of type Ii error should not exceed 0.10

Hence