A binomial probability distribution has p = .20 and n =...

A binomial probability distribution has $p = .20$ and $n = 100$. If required, round your answers to four decimal places. Use \"Continuity correction factor, if necessary\". Use Table 1 in Appendix B. a. What are the mean and standard deviation? $\mu =$ $\sigma=$ b. Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Select Explain. Select c. What is the probability of exactly 24 successes? d. What is the probability of 18 to 22 successes? e. What is the probability of 14 or fewer successes?

Transcript

00:01 This problem we're told that a binomial probability distribution has values of p equaling 0 .2 and n equaling 100.

00:12 Now we know that q in a binomial distribution is always equal to 1 minus p so 1 minus 0 .2 would give us a q value of 0 .8.

00:22 Let's see what we're asked to find here.

00:24 So in part a we're first asked to find the mean and standard deviation.

00:30 So you could see that the the equation up here is equal to n times p so our n is 100 and we'll multiply that by our p which is 0 .2 and that will give us 20 for our mean.

00:47 Our standard deviation then is equal to square root of n times p times q square root of n times p times q.

01:01 So we will take the square root of 100 times 0 .2 times 0 .8 plug that into my calculator real quick here and come out with 4.

01:14 So here we have our mean and standard deviation for this distribution.

01:21 Here in part b it says is this situation one in which the probability binomial probability can be approximated by the normal distribution.

01:29 Yes or no and explain.

01:31 So in order to decide that we have to multiply n times p and then n times q.

01:36 If n times p is greater than or equal to 5 and n times q is greater than or equal to 5 then we are able to approximate it to the normal.

01:46 So n times p we already found up here for the mean that was 20 so that's good but let's now do n times q.

01:56 So 100 times 0 .8 gives us 80 so that is also good.

02:01 So the answer is yes we're able to approximate to normal.

02:06 Now in part c we're asked to find the probability of exactly 21 successes.

02:11 So we're finding probability of x equaling 21.

02:16 Now we can't find an equal sign in our distribution our normal distribution so we'll have to do a continuity correction meaning we're just going to subtract and add 0 .5 to 21.

02:27 So what we're actually finding is the probability of x being between 20 .5 and 21 .5.

02:37 Now what we need to do is find z scores for both of these numbers using the red z equation i have up here on the screen.

02:43 Start with 20 .5 using the equation z is equal to x just 20 .5 minus the mean which is 20 divided by the standard deviation which is 4.

02:59 If we type that into our calculator round to two decimal places come out with 0 .13.

03:06 Now for 21 .5 that's x subtract the mean and divide by the standard deviation again we'll plug that into our calculator round to two places decimal places and we come out with 0 .38.

03:23 Now that we have those two z scores let's find them in our z table on the left side of the screen.

03:29 So 0 .13 gives us 0 .5517...

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