"Simulation Simulate drawing 100 simple random samples of size...

Question

"Simulation Simulate drawing 100 simple random samples of size n = 40 from a population whose proportion is 0.3.

d. How do we know that a rejection of the null hypothesis results in making a Type I error in this situation?"

Accepted Answer

Welcome back, everyone. In this problem, you draw 150 simple random samples of size N equals 30 from a population with true proportion of P equals 0.2. You perform a one-sided test that says the another hypothesis is that the proportion equals 0.2 versus the alternate hypothesis that the proportion is greater than 0.2 at a significance level alpha of 0.01. Which description correctly identifies a type 1 error in this simulation and what is the approximate number of such errors expected among the 150 samples? A says rejecting the null hypothesis is a type one error only if the sample proportion is less than 0.2, and we can expect about 148.5 such errors. B says rejecting the null hypothesis cannot be a type 1 error in a one-sided test, and we should expect zero Type 1 errors. CE says rejecting the null hypothesis for a sim simulated sample when data were generated with P equals 0.2 is a type one error, and we can expect about 1.5 such errors. And the DU says that rejecting the null hypothesis is a type 2 error in this one-sided test and we can expect about 1.5 such errors. Now, we know that our another hypothesis, OK, is that our proportion is 0.2 and our data mechanism says that each sample is generated with a true proportion. P of 0.2, OK. Now what do we know about type 1 errors? Recall that a type one error. Basically says. That There's a or it's basically the probability that we reject the null hypothesis even when the null hypothesis is true, OK? So that's what we need to check. We need to check what that possibility would be. Now, using the sample proportion test statistic formula. We know that the Z value then is going to be equal to the sample proportion minus 0.2. Divided by the square root, OK, of our true proportion, 0.2, OK, multiplied by 1 minus or proportion. So that's 1 minus 0.2 or 0.8, divided by our sample size N, which is 30. So now, if we're using the one-sided test. OK, using the one-sided test. For our significance level alpha of 0.01 and our critical Z value. OK. So our critical Z value is Z 99 at a 99% confidence, so which is gonna be 2.326, OK. Then we're going to reject basically. When Our sample proportion minus 0.2 is greater than 2.3232.326, my apologies, is multiplying the square root of 0.2 multiplied by 0.8 divided by 30. So for us to figure out what our sample proportion would be in this case, what it would have to be greater than, let us evaluate our standard error to get the critical cutoff. Now, in this case, 0.2 multiplied by 0.8 is 0.16, and when we find a square root of that divided by 30, It would be approximately well let me write it here. It would be approximately 0.073027, OK? So Our sample proportion minus 0.2 would be greater than this value, which when we multiply, that means it's going to be greater than approximately 0.16973, which means that we're going to reject if the sample proportion equals 0, greater than 0.2 plus. 0.16973. Or if it is greater than 0.36973, approximately. Now, what does this mean? Well, since the data are generated under the no hypothesis, basically, any sample with a sample proportion greater than 0.36973, OK. So, any sample. With our sample proportion greater than 0.36973. Use a rejection. OK, while or no hypothesis is true, a type one error, OK? That's a type one error. And in that case, We can expect. As we can expect about 150 multiplied by 0.01 or 1.5 such errors, since our significance level is 0.01. So that's roughly 1 or 2 such errors. So now we know that this is what it means. Let's look at our answer choices to see which one of these descriptions would correctly identify a type one error and the approximate number of such errors. When we go back to answer twice A, it says that rejecting our null hypothesis is a type one error only if the sample proportion is less than 0.2, and we can expect about 148.5 such errors. But this is incorrect because type one error is re rejecting when the null is true regardless of whether the sample proportion is less or greater than 0.2. And the numerical value is not the expected count of type one errors. Therefore, A is incorrect. In answer choice B, it says that rejecting a null hypothesis cannot be a type 1 error in a one-sided test, but one-sided tests still have a non-zero type 1 error rate equal to our significance level alpha. Here, alpha equals 0.01, so some type 1 errors are expected. We, we can be 100% sure that we won't have a type 1 error. Therefore, B is also incorrect. She says rejecting another hypothesis for a simulated sample when data were generated with P equals 0.2 is a type 1 error, and we can expect about 1.5 such errors. This is our correct statement. This is what we were saying earlier, OK? So, therefore, C is the correct answer. And we can be sure we're right, because indeed, it says rejecting the null hypothesis is a type 2 error in this one-sided test. But a type 2 error is failing to reject when the null false. Rejecting when the null is true are type 1 errors, not type 2. So C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

"Simulation Simulate drawing 100 simple random samples of size n = 40 from a population whose proportion is 0.3.
d. How do we know that a rejection of the null hypothesis results in making a Type I error in this situation?"

Key Concepts

Type I Error

Null Hypothesis in Hypothesis Testing

Sampling Distribution and Random Sampling

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