A researcher believes that about 71 of the seeds planted...

A researcher believes that about 71% of the seeds planted with the aid of a new chemical fertilizer will germinate. He chooses a random sample of 155 seeds and plants them with the aid of the fertilizer. Assuming his belief to be true, approximate the probability that fewer than 112 of the 155 seeds will germinate. Use the normal approximation to the binomial with a correction for continuity. Round your answer to at least three decimal places. Do not round any intermediate steps. (If necessary, consult a list of formulas.)

Transcript

00:01 Hi, i'm david and i'm here to help you answer your question.

00:03 Now let me bring up your question here.

00:06 In the question here, we are going to discuss about the binomial approximation to the normal.

00:12 Let me remind you that it will the x that followed by the binomial with the n and the p.

00:18 And then x can be approximated by the normal with the mean it will equal to the n times b.

00:30 Standard deviation equal to the square root of the n times b times 1 minus b and because it followed by the normal it will turn the x will minus the mean over the standard deviation and then we obtain the standard normal in the equation we're given the 71 % of the seed planted with the aid of the new chemical fertilizer will germinate so means then the b equal to the zb171 we have the sample and equal to the 155.

01:03 So from here, if we call the x equal to the number of the seat will terminate and then x will follow by the binomial with the n equal to the 155 p equal to the 171.

01:23 Now the question asks let you find the probability that fewer than 112 on the seat, so x will be smaller than the 112.

01:35 12 and now want to find this probability because x here we can approximate x by the normal with the mean equal to n times b so n equal to 155 times b will be the 171 yet equal to 155 times 0 .71 equal to the 110 .05 and the standard division equal to the square root of the 150 times 0 .71 times 1 minus upon 71.

02:12 And then we get equal to 0 .29 taking the square root will be the 5 .649 and therefore we can convert the x into the normal by using the formula here.

02:31 Here because we have this one will be the strictly less than the 102 set of and for this one, we no need to use the continuity correction.

02:41 So therefore, when you conversion into the z, so we will have, we check, you think the 112 minus the mean, it will be 110 .05.

02:53 We will divide them by the standard deviation.

02:57 And if we compute, we will have the z will be smaller than we have the 112 minus 1 1 1 05, divided by the the answer equal to the zobon 3, 4, 5.

03:11 Now to find this probability, we will need to bring up the z table...

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