 Hi, this is Dr. Don. I had a student ask a question about how to solve a hypothesis test about a proportion. And this is an example of that type of problem. We're told that a researcher claims that at least 46 percent of adults in a certain country think the government is not aggressive enough in a random sample of 600 adults in that country 41 percent say the government is not aggressive enough. And we want to know at alpha point of one is there enough evidence reject the center's claim? The center's claim was at least. Remember in the table that I shared before the at least is a greater than or equal to situation. And since it's a form of equality that means that this claim has to be the null. And we're looking at that the claim is at least 46 percent which is what it says here. And saying it in mathematical form we've got the population parameter p is greater than or equal to 0.46 and the alternative is p less than 0.46 because this is a left tail the operator points to the left we should have a negative value of our critical value of z. And by the way for the proportion test we always use the z distribution. And this question wants us to find that critical value and also identify the rejection region find the standardized test statistic then come up to a decision and a conclusion. So the way that I would recommend people do this is to use the calculator that I posted on my website. Let me show you that. Okay I've brought up my website which is just drdonright.com whenever you go there it may look a little different obviously since I have different posts probably on the front page. But you can just click on the business 233 link up there and it will bring up these excel calculators that I've been making and we want the single sample z test for proportion. And so I'm going to click on that and bring it up. I've got some instructions there and after a minute or so this spreadsheet should open up. Now you can do your work directly into this this window that I have here or if you want you can click down here and expand it into a full-size workbook. But for this problem we're just going to work it directly in this window. And I would note that down at the bottom there's a scroll bar you can scroll over and you can see because this is a test about a proportion we're going to use the normal approximation if possible which allows us to use the z distribution and we've got to check that n and p are both greater than five. And here once you enter your data this calculator will automatically check that for you and if it is okay to proceed it'll be green here okay to proceed if it's not okay to proceed it will turn red. So we need to enter in our population mean which in this case was 0.46 you only enter data in the blue cells the yellow and green and orange cells are all protected so you shouldn't be able to change anything. The sample state of deviation is being calculated for you but you've got to enter the sample proportion which in this problem was 0.41. Again remember although it said 41% you have to use decibels in these calculators and instead crunch as well. The scientific level was alpha sample size was 600. The next thing we need to do is select the claim math operator remember that in this case it said that the the tax center says at least 46% so that's the greater than or equal to change that you just click on it and then select the right operator and we select the greater than or equal and that tells us automatically that the claim is the null. We know that restating it in math formalization we've got that p is greater than 0.04 0.46 the alternative is p less than 0.046 the calculator gives you that it tells you it's the left tail test which again because it's pointing to the left that is left tail that means our critical value of z is going to be negative always for left tail test z just depends upon the tail of the test and the alpha and so we know that it's minus 2.33 if we ran that two decimal places rejection region would be any test statistic that is less than to the left of minus 2.326 the test statistic which is used found using our standard formula is minus 2.4574 which is to the left of minus 2.326 and down here it tells you is z in a rejection region yes alpha is calculated I'm sorry p is calculated for you and is less than our alpha which it tells you that as well so down at the bottom we get our decision reject the null and the conclusion at the one percent significance level there is enough evidence to reject the claim which was the null and again we found that out up here and I'm also telling you that at the bottom we reject the null because z is in the rejection region we reject the null because p is less than alpha so I hope this helps