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The numbor successes and the sample simple random sampie Irom population are given below: 30, Ho 04, Ha" P+0.4 0,01 Determino the sample proportion; Decide who...

Question

The numbor successes and the sample simple random sampie Irom population are given below: 30, Ho 04, Ha" P+0.4 0,01 Determino the sample proportion; Decide whother using the ono-proportion z-tost is appropriate appropriate , tw one-propontion z-test perform the specifed hypothesis test Click here viow table ofareas under the_standard normal curvu for negalive Click hore viewy Lable ol areas Undlel walues olz Iho standard noma cunetot posilivu values 02 Idontity Ihe cribcal value(s}, appropr

The numbor successes and the sample simple random sampie Irom population are given below: 30, Ho 04, Ha" P+0.4 0,01 Determino the sample proportion; Decide whother using the ono-proportion z-tost is appropriate appropriate , tw one-propontion z-test perform the specifed hypothesis test Click here viow table ofareas under the_standard normal curvu for negalive Click hore viewy Lable ol areas Undlel walues olz Iho standard noma cunetot posilivu values 02 Idontity Ihe cribcal value(s}, approprato Select Ihe corroct choico below and, 'necossary the answur box compluld voui HneUmr 17u/2 (Round two doclmal placos nuodud ) (Round twa ducimm plucus needud ) Zu= (Round two decimal places as needed ) The ono-proportion z-lest approprate



Answers

We have given the number of successes and the sample size for a simple random sample from a population. In each case, do the following. a. Determine the sample proportion. b. Decide whether using the one-proportion z-test is appropriate. c. If appropriate, use the one-proportion z-test to perform the specified hypothesis test. $$x=10, n=40, H_{0}: p=0.3, H_{\mathrm{a}}: p < 0.3, \alpha=0.05$$

Yeah were given some hypothetical data and we want to determine if we can use the one proportion Z test to run a hypothesis test. The first thing that we need is our sample proportion which is eight out of 40 Which is 0.2 as a decimal. Now our technical conditions, these three things have to be met. We have to have a simple random sample and the problem says that we did get them from a simple random sample and then our sample size times our population proportion gas of 0.3 should be greater than five and 40 times 400.3 is greater than five and then sample size of 40 times one minus 10.3 is also greater than or equal to five. So it is appropriate to use the one prop z test. Now as we start our Hypothesis tests we've got are null and alternative hypotheses given to us. And then for our Z score we're going to use this formula. We already know our sample proportion is .2 P zero is 3.3. And then I'm just gonna plug in .31 -3 and sample size of 40. And then when we run this calculation you should get a Z. score of negative 1.38. And then when you look that up in your standard normal probability table in the back of your book You will get a P value of .084. And now we have to decide if we are rejecting or failing to reject the null hypothesis. Well, they've told us we're using a significance level of 0.1 and r p value is less than 0.1. So we will reject H. O. And say our population proportion is less than .3.

Is it appropriate to conduct the one proportion Z test on this particular set of sample data? First of all, our sample proportion is going to be three out of 100 Which is 0.03 as a decimal. Now for our criteria that have to be met, is this a simple random sample? Yes, it is. Does the sample size of 100 times are hypothesized. Population proportion of 0.4 come out greater than or equal to five 100 times .04 is only equal to four. So this is not map now. The second criteria here, 100 times one minus 10.4 is going to be 96 so that is definitely bigger than five. However, because the sample size times a hypothesized population proportion is not bigger than or equal to five, it is not appropriate to carry out the one proportion Z test and we will stop right there.

Are we able to use the one proportion Z test on this set of sample data? RP half our sample proportion is 40 out of 50, Which is 0.8. And then our criteria that have to be met. Is it a simple random sample size? Yes it is. They tell us that. Does our sample size of 50 multiplied by our hypothesis. Population proportion of .7. Is that bigger than five? Yes. And is 50 times one minus 10.7. Also bigger than or equal to five? Yes, it is. So it is appropriate to use the one proportion see test. So let's do it. Are null and alternative hypotheses are given notice the H O. Is a two tailed tests and not equal to here. So that's going to influence what we do. When we calculate our P value R Z score first of all is going to be our sample proportion minus are hypothesized population proportion Over the square root of .7 times 1 -1 over a sample size of 20. And running this calculation we get a Z score of .98. Now when I look up .98 in the standard normal probability table, it gives us a probability of 0.83 65 Now I'm going to do one minus 10.8365 to give us the low end probability of our bell curve which is 0.1635 But Because this is a two tailed test we have to multiply that value by two Which will give us about .3 27 Now, are we going to reject H. O. Or failed to reject H Oh well this significance level of 0.5 R p value is way larger than that. So we are going to fail to reject H. O. Mhm And conclude that our population proportion is .7.

Is it appropriate to run the one proportion Z test with the given sample data? Well, the first thing we're gonna need to know is our sample proportion, which in this case is 40 out of 50 Or 0.8. And then our technical conditions here have to be met to run the one proportion Z test. Is it a simple random sample? Yes, they tell us that is our sample size times are hypothesized. Population proportion bigger than 5, 50 times 500.6 is bigger than five and 50 times one minus 10.6 is also bigger than five. So it is appropriate to use the one proportion Z test. So let's do it. No. An alternative hypotheses are given and now we need our Z score. Our sample proportion was 0.8. Yeah. Are hypothesized, population proportion is .6 and that's going to go here in a few places And our sample size was 50. And after you run this calculation we get a Z score of 2.89. Now we need that to give us our p value from our table in the back of the book. The standard normal probability table When we look up 2.89 gives us a value of .9981. But because where are alternative hypothesis is greater than 0.6, we need to do 1 -1981 to give us our p value, which is .0019. Now we have to decide are we going to reject H. O. Or fail to reject H. O. While looking at a significance level of 00.1 This p value is definitely smaller than 0.1 So we are going to reject H. O. Which means that we are going to now believe the population proportion is bigger than .6.


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