Question

10) (1 point) A manufacturer of electronic kits has found that the mean time required for novices to assemble its new circuit tester is 2.7 hours, with a standard deviation of 0.8 hours. A consultant has developed a new instructional booklet intended to reduce the time an inexperienced kit builder will need to assemble the device and the manufacturer needs to decide whether or not to send out the new booklet.

The testable hypotheses in this situation are H0:μ=2.7H0:μ=2.7 vs HA:μ<2.7HA:μ<2.7.

1. Identify the consequences of making a Type I error.
A. The manufacturer sends out a helpful instructional booklet.
B. The manufacturer does not send out a helpful instructional booklet.
C. The manufacturer sends out an unhelpful instructional booklet.
D. The manufacturer does not send out an unhelpful instructional booklet.

2. Identify the consequences of making a Type II error.
A. The manufacturer sends out an unhelpful instructional booklet.
B. The manufacturer does not send out a helpful instructional booklet.
C. The manufacturer sends out a helpful instructional booklet.
D. The manufacturer does not send out an unhelpful instructional booklet.

To monitor the assembly time of inexperienced kit builders using the booklet, the manufacturer is going to take a random sample of 10 novices and calculate the mean time to assemble the circuit tester. If it is less than 2.6, they will send out the new instructional booklet. Assume the population standard deviation is 0.8 hours.

3. What is the probability that the manufacturer will make a Type I error using this decision rule? Round your answer to four decimal places.

4. Using this decision rule, what is the power of the test if the actual mean time to assemble the circuit tester is 2.1 hours? That is, what is the probability they will reject H0H0 when the actual average time is 2.1 hours? Round your answer to four decimal places.

(Please answer number 4)

Answer 1

1. Identify the consequences of making a Type I error.

Answer: C. The manufacturer sends out an unhelpful instructional booklet.

2. Identify the consequences of making a Type II error.

Answer: B. The manufacturer does not send out a helpful instructional booklet.

3. Calculating the type 1 error.

A type I error occurs when one rejects the null hypothesis when it is true which is also known as alpha.

so alpha = P(rejecting null hypothesis/ H0 is true)

= P(xbar<2.6 / mu = 2.7)  

{mu = 2.7 and st. error = 0.8/ sqrt(10) = 0.25}

= P (z<( 2.6 - 2.7) /0.25) = P(z< -4) = 0.00003

4. We know that,

Power = 1- beta = - type 2 error

A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. The probability of a type II error is denoted by *beta*.

beta =  P(accepting null hypothesis/ H1 is true)

= P(xbar>2.7 / mu = 2.1)  

{mu = 2.1 and st. error = 0.8/ sqrt(10) = 0.25}

= P (z>( 2.7 - 2.1) /0.25) = P(z>2.4) = 0.0082

So, Power = 1- beta = 0.9918 = 99.18%