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You wish to test the follawing claim (H.) at significance level of a = 0.001.Ho:P1 = P2 Ha:Pi < P2You obtain 92.5% successes in a sample of size n1 =318 from the...

Question

You wish to test the follawing claim (H.) at significance level of a = 0.001.Ho:P1 = P2 Ha:Pi < P2You obtain 92.5% successes in a sample of size n1 =318 from the first population. You obtain 98.1% successes in a sample of size n2 683 from the second population: For this test, you should NOT use the continuity correction_ and you should use the normal distribution as an approximation for the binomial distribution_What is the test statistic for this sample? (Report answer accurate to three deci

You wish to test the follawing claim (H.) at significance level of a = 0.001. Ho:P1 = P2 Ha:Pi < P2 You obtain 92.5% successes in a sample of size n1 =318 from the first population. You obtain 98.1% successes in a sample of size n2 683 from the second population: For this test, you should NOT use the continuity correction_ and you should use the normal distribution as an approximation for the binomial distribution_ What is the test statistic for this sample? (Report answer accurate to three decimal places:) test statistic What is the p-value for this sample? (Report answer accurate to four decimal places:_ p-value



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. $$x=3, n=100, H_{0}: p=0.04, H_{\mathrm{a}}: p \neq 0.04, \alpha=0.10$$

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.

Can we use the hypothesized data given to run a one proportions Z test? Well, first thing we need to know is our sample proportion which is 10 out of 40, Which is 0.25 as a decimal. Now the technical conditions here have to be met to run the tests. So we need a simple random sample, they tell us that we have that and then our sample size 40 times are hypothesized. Population proportion of 400.3 has to be bigger than five. Well it is 12, so it is bigger and then 40 times one minus 10.3 has to be bigger than or equal to five. And that is also met. So it is appropriate to use the one proportion Z test. Now we're going to conduct that test Are now an alternative alternative hypotheses are given. So now we have to figure out our Z score. Well, our sample proportion we already calculated as .25 Are hypothesized population proportion is .3 And our sample size is 40. And when you run this calculation we could get a Z score of negative 0.69. Now we'll go to your standard normal probabilities chart and look up that Z score of negative 0.69 And that gives us a p value of 0.2451 Now we have decided to reject or failed to reject H. L. And we're going to base that off of a significance level of 0.05. Well .2451 is definitely bigger than .05. So we are going to fail to reject h. L.. And continue to believe that the population proportion is equal 2.3.

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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