A consumer group plans to test whether a new passenger car...

A consumer group plans to test whether a new passenger car that is advertised to have mean highway miles per gallon of at least 33 actually meets this level. They plan to test the hypothesis using a significance level of 0.05 and a sample size of n = 100 cars. It is believed that the population standard deviation is 3 mpg. Based upon this information, if the "true" population mean is 32.0 mpg, what is the probability that the test will lead the consumer group to "accept" the claimed mileage for this car? A) About 0.45 B) Approximately 0.0455 C) About 0.9545 D) None of the above

Transcript

0:00 Hello everyone.

00:01 So in this question we have given three alternating series and we have to check which alternating series can be used to show convergence using alternating series test.

00:11 So now let us understand first that what is alternating series test? alternating series test.

00:25 So the alternating series test says that if limit n -tenthens.

00:33 To infinity absolute value of a .n.

00:37 Is equal to zero and series is decreasing, that is, absolute value of a .n plus one should be less than equal to absolute value of a .n.

00:47 So if these two conditions satisfies by alternating series, then we can say that alternating series is a convergent.

00:57 So now we will answer our question one by one.

00:59 So our first question is, 4 minus 1 divided by 9 plus 1 minus 1 divided by 81 plus 1 divided by 4 minus 1 divided by 729 plus so on plus 8 divided by 2 power n minus 1 divided by 3 power n plus 1 and so on.

01:26 So this is our alternating series.

01:28 So now in this series, here we can write absolute value of a1, that is first term, is equal to 4, then absolute value of a2 is equal to 1 divided by 9, then absolute value of a3 is equal to 1, absolute value of a4 from the series is equal to 1 divided by 81.

01:54 Similarly, absolute value of a5 is equal to 1 divided by 4 and so on.

02:02 So now as we can see here that absolute value of a1 is greater than absolute value of a2.

02:11 Then absolute value of a2 is less than absolute value of a3.

02:18 Then absolute value of a3 is greater than absolute value of a4.

02:25 Then absolute value of a4 is less than absolute value of a5.

02:33 So therefore we can say that absolute value of an is not monotonically decreasing.

02:51 And if one condition fails, then no need to find other condition.

02:56 So therefore, we can say that...