probability of making a type 1 error, and that probability is 0.05. So there's 5% chance that you might be committing a type 1 error with your hypotheses testing. Anyway, then we need to write down our formula, it's a z-test because we know the population standard deviation. So the formula is sample mean minus the hypothesized mean, divided by standard deviation over the square rootof the sample size. Now the problem tells that we randomly picked 100 bottles and the mean for these bottles was 19.55. And then my claim is 20, now in the numerator I'm looking at the difference between the actual number and the claimed number. But this, alone, does not tell me a lot of things. I need to standardize this difference, right? Because if we are talking about a Large scale, this difference might be maybe too little. If we're talking about a small scale, then the same difference might be relatively large. So, we need to make this difference of scale independent. And this is why we need to standardize this difference. So the Standard deviation for this problem was two and then the number of models in our sample is, 100. And if I solve this equation, on the numerator, I have -0.45, and here in the denominator, I have 2 over 10. So what does it give me? Well, this gives me, if you do the math, -2.25. This is the z score that I calculated. Now, once I have a z score, you already know what to do with it. We need to look up the z table, so let's go ahead and do that. So, this is my z table, and my calculated z value or z score was -2.25. So I find that negative 2.2 is here. This is 2.20, 2.21, 2.22, 2.23, 2.24, and negative 2.25, okay? And I see that the probability value here is 0.0122. So I found the probability value of 0.0122. What does it mean? Let's have a look at this distribution, okay? So we have this mean value of 20, the hypothesized mean value. And then we have This value of 19.55. Now, if you look at the alternative hypothesis, the sign on the alternative hypothesis tells that this should be a left-tailed test because it's pointing to the left. So I am interested in the left side of the distribution. This is a symmetrical distribution, the centre is the mean and I'm interested in the left half of this distribution. Now, the z value of -2.25, again, if you remember from a previous video, gave us an area of 0.0122. So what does it mean? I'm gonna change color here. So it means that the area under the curve to the left of this z value is 0.0122. So this area here is 0.0122. The area to the left of 19.55 is 10.0122 because -2.25 is the z score that corresponds to 19.55 for this particular distribution. So this means that Picking this particular sample, which gave me a sample mean of 19.55, more and more extreme sample, meaning a sample that gives me a mean to the left of this particular value has a probability of 0.0122. And essentially, the P value is 0.0122, this is what I'm looking for. Once again, right? So there are these bottles, okay? If the null hypothesis is true and that the mean for this population is greater than or equal to 20, you ask yourself, if this is correct, what are the chances I would pick a sample that would give me this particular mean, or a more extreme mean, a mean that is on the left of this point. That is more, it's further from this point. You ask
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- Fall '15
- Null hypothesis, Statistical hypothesis testing