When a thread cutting machine is operating properly only 2...

When a thread-cutting machine is operating properly, only 2% of the units produced are defective. Since the machine was bumped by a forklift truck, however, the quality-control manager is concerned that it may require expensive downtime and a readjustment. The manager would like to set up a right-tailed test at the 0.01 level of significance to identify the proportion of defectives using a sample of 400 units produced. (a) If 12 of the 400 units produced are defective, what conclusion would the quality-control manager make? (b) Find the probability of type II error if the actual population proportion of defectives is (i) 0.02? (ii) 0.03? (iii) 0.04? (iv) 0.05? (v) 0.06?

Transcript

00:01 Hello students, here in the given problem first here given the study conclusion here.

00:06 So first study hypothesis here h naught is p is equal to 0 .02 that is the machine is operating properly and the proportion of the e defective is 0 .02 and alternative hypothesis p is greater than 0 .02 that is the machine is not operating properly and the proportion of the defective is greater than 0 .02.

00:32 So here given n is equal to 400 and find the sample proportion that is 12 divided by 400 so this is a 0 .03 and here alpha is 0 .01.

00:46 Now state the state statistic here the formula is z is equal to p hat minus p divided by square root of p into 1 minus p divided by n.

01:03 So here p hat is 0 .03 minus p is 0 .02 divided by square root of 0 .02 into 1 minus 0 .02 divided by 400.

01:17 So here z is 1 .4285.

01:24 Now find the critical value.

01:27 Here critical value for the p -test for alpha is 0 .01 the value is 0 .33.

01:34 So here we can say that z calculated is less than z critical value.

01:41 So we can say that we fail to reject h naught.

01:51 Now take the conclusion here conclusion is we fail to reject h naught means h naught is expected.

02:04 So here the machine is operating properly and the proportion of defective is 0 .02.

02:29 Now we part to find the type 2 error for each of the cases.

02:39 The first case where p is equal to 0 .02.

02:44 So here 0 .02 therefore the beta is equal to the probability of z which is less than equal to to find the above part 0 .02 the proportion is 1 .43.

02:59 Therefore the answer is 0 .9236.

03:05 So for the p is equal to 0 .02 the type 2 error is 0 .9236.

03:13 Now the second one for the p is equal to 0 .03.

03:20 So here first find out the z test 0 .03 minus 0 .03 divided by square root of 0 .03 into 1 minus 0 .03 divided by 400.

03:35 So this value is 0.

03:37 Therefore the probability of z is less than equal to 0 which is 0 .5.

03:43 Therefore in that case the beta is 0 .5.

03:49 Now third part here for the p is equal to 0 .04...