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In order t0 conduct hypothesis test for the population proportion, you sample 320 observations that result In 128 successes: (You May IInd useful to reference the a...

Question

In order t0 conduct hypothesis test for the population proportion, you sample 320 observations that result In 128 successes: (You May IInd useful to reference the appropriate table: z lable or tuable)45; HA:0.45.Calculate the value of the test statistic: (Negatlve value should be Indicated by minus sign. Round intermedlate calculatlora et least " decimal places and final answer to decimal places )Keenc02. Find the p-value.Tanduue001ocit puolue 0.025Unpb eeweua 005Uue Keoeld O0Oalp 3062

In order t0 conduct hypothesis test for the population proportion, you sample 320 observations that result In 128 successes: (You May IInd useful to reference the appropriate table: z lable or tuable) 45; HA: 0.45. Calculate the value of the test statistic: (Negatlve value should be Indicated by minus sign. Round intermedlate calculatlora et least " decimal places and final answer to decimal places ) Keenc 02. Find the p-value. Tanduue 001 ocit puolue 0.025 Unpb eeweua 005 Uue Keoeld O0 Oalp 3062



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. $$x=40, n=50, H_{0}: p=0.6, H_{\mathrm{a}}: p > 0.6, \alpha=0.01$$

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.

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.

Mhm. In this problem, we're gonna be conducting a two proportion Z test. The first thing we need to do is figure out our sample proportions from Group one. The sample proportion is 18 out of 40 which is equal to 400.45. Group to sample proportion is 30 out of 40 or 75 And our combined pooled sample proportion would be 48 successes out of 80 which would give us 0.6. Now, before we actually do H. O and H. A, we should check that our technical conditions are met. The problem told us we have simple random samples and they are independent. And then our number of successes for both groups has to be bigger than 5, 18 and 30. Check and then our sample size minus the number of successes for each group has to be bigger than five. Well, and Group 1 40 takeaway 18 is obviously bigger than five. And a group to 40 take away 30 is 10 and that's bigger than five. So we are good to go here for our null hypothesis. Mhm. Yeah. Population proportion number one is equal to population proportion number two. And for alternative hypothesis, population proportion one is less than population proportion number two. Okay. Yeah. And now I've plugged in our sample statistics into our test statistic formula and we should get a Z score value of negative 2.74 And when we reference that in the appendix, in the back of the book, Z score of negative 2.74 corresponds to a p value of 0.31 which is smaller than our significance level. So we are going to reject H. L. So we have enough evidence to claim that population proportion number one is less than population proportion number two. Now the last part of the question at 80% confidence interval. I've plugged in our test statistics here. Um, one thing to know. RZ for our test statistic for 80% is going to be 1.28 when we multiply that by the big square root part will get a margin of error of 1.33 that gets added and subtracted from negative 0.3. And we get an interval from negative 0.4332 negative 0.167 And this is we are 80% confident. The difference in population proportions is somewhere in this interval.


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