Parks and Mental Health In Exercises 13–18, use the figure, wh...

Question

Parks and Mental Health In Exercises 13–18, use the figure, which shows the percentages from a survey of two hundred 18- to 24-year-olds in the United States who say that various park and recreation activities have a positive impact on their mental health. (Adapted from National Recreation and Park Association)

Socializing and Taking Classes At α=0.05, can you support the claim that the proportion of 18- to 24-year-olds who benefit mentally from socializing in parks is different from the proportion who benefit mentally from taking classes in parks?

Accepted Answer

Hello there. Today we're gonna solve the following practice prom together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A national education report compared the proportion of students who graduate within 4 years at public and private universities. A random sample of 1000 students from public universities showed that 540 graduated within 4 years. A random sample of 1000 students from private universities showed that 610 graduated within 4 years. At alpha equals 0.05, can you support the claim that the proportion of students graduating within four years is different between public and private universities? Awesome. So it appears for this particular problem we're trying to determine whether or not we can support the claim. That the proportion of students graduating within 4 years is different between public and private universities given a level of significance alpha that is equal to 0.05. So now that we know, we're ultimately trying to determine whether or not we could support this specific claim, and that's our final answer we're trying to solve for. Let's take a moment to read off our multiple choice answers to see what our final answer might be. A is there is sufficient evidence to support that the graduation rate is different between public and private universities. B is there is insufficient evidence to support that the graduation rate in private universities is higher than in public universities. C is there is sufficient evidence to conclude that the graduation rate is the same in both types of universities, and D is there is insufficient evidence to support any difference in graduation rates between public and private universities. Awesome. So our first step in order to solve this problem is we need to determine our hypotheses. So let us note that we will use P subscript 1, P1 to represent the proportion of public university students who graduate within 4 years. And we will use P2 to denote the proportion of private university students who graduate within 4 years. So with that in mind, we can now determine our null and our alternative hypotheses. So the null hypothesis H0 is that P1 will equal P2 and our alternative hypothesis will state that P1 cannot equal P2. Fantastic. That's our first step. Our second step now is we need to determine the pooled proportion. Which, as you should recall, Phat is equal to X1 plus X2 divided by N1 plus N2, which when we plug in our known variables, we will find that this is equal to 540 plus 610, all divided by 1000 plus 1000. And when we plug in this expression into our calculator and we round to three decimal places, we will find that it's equal to 0.575. Awesome. So, moving right along here, our 3rd step now is we need to determine, or rather, we need to solve for the standard error. Which will denote this as SE. So as we should recall, the equation to solve for the standard error is that SE is equal to the square root of. Phat multiplied by parentheses 1 minus Phat multiplied by parentheses 1 divided by N1 plus 1 divided by N2. Once again in parentheses, which this is equal to when we plug in our known variables. This will be equal to the square root of 0.575, multiplied by 0.425 multiplied by parentheses 1 divided by 1000 plus 1 divided by 1000. And when we plug in this expression into our calculator and we round 2. Approximately 123455 desal places, we will find that our standard error is approximately equal to 0.02211, which once again, this is when we round to 4, or rather 5 decimal places. Fantastic. Moving right along here, our fourth step now is we need to compute our Z statistic, which, as we should recall, the Z statistic Z is equal to Phat 1 minus P 2 divided by SE, which is equal to when we plug in our known variables, 0.54 minus 0.61. Divided by 0.02211, which when we plug that expression into our calculator and we round to three decimal places, we will find that it's approximately equal to -3.166. Fantastic. So, moving right along here. Our 5th step now is we need to determine our decision rule. And as you should recall, in order to solve for the decision rule, we need to note that for a level of significance, alpha equals 0.05, when we're dealing with a two-tailed critical Z value. So, once again, Let's take a moment here to pause. So we're told the level of significance by the prom itself, and based on everything we've discovered so far, we need to recognize that we're dealing with a two-tailed test. And because we're given this specific value for the level of significance and we know that it's a two-tail test that we're dealing with, we can determine that from using our table in our textbook that the critical Z value. Values are plus or minus 1.96, and once again, these are critical values, which we'll call CV for short. So, plus or minus 1.96 are our critical values. So that will mean that the absolute value of Z is equal to 3.166, which is greater than 1.96. And because the absolute value of Z is equal to 3.166 and this value is greater than 1.96, we can reject the null hypothesis. So I put an X next to H0 to remind us, so we can reject the null hypothesis, and because we can reject the null hypothesis. We can make the following conclusions. So at the alpha equals 0.05 level, there is sufficient evidence to support the claim that the proportion of students graduating within four years is different between public and private universities. And this corresponds with multiple choice answer letter A, which once again is there is sufficient evidence to support that the graduation rate is different between public and private universities. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Proportion

Hypothesis Testing

Significance Level (α)

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