Coin factories A and B produce coins for which the...

Coin factories A and B produce coins for which the probability of heads $p$ is a betadistributed random variable. Factory A has parameters $a=b=10$, and factory B has $a=b=5$. (a) Design a hypothesis test for $\alpha=5 \%$ to determine whether a batch is from factory A . (b) What is the probability of detecting factory B coins? Hint: Use the Octave function beta_inv. Assume that the probability of heads in the batch can be determined accurately.

Coin factories A and B produce coins for which the probability of heads $p$ is a betadistributed random variable. Factory A has parameters $a=b=10$, and factory B has $a=b=5$.
(a) Design a hypothesis test for $\alpha=5 \%$ to determine whether a batch is from factory A .
(b) What is the probability of detecting factory B coins? Hint: Use the Octave function beta_inv. Assume that the probability of heads in the batch can be determined accurately.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In the context of the problem, it involves setting up a null hypothesis (that the batch comes from a specified factory, for instance, factory A) and an alternative hypothesis, then determining whether observed outcomes (based on the probability of heads measured) provide sufficient evidence to reject the null hypothesis at a given significance level.

Significance Level

The significance level, typically denoted by ?, is the probability of rejecting the null hypothesis when it is actually true (a Type I error). In this problem, a 5% significance level means there is a 5% risk of mistakenly concluding that a batch did not come from factory A when it actually did.

Beta Distribution

The beta distribution is a family of continuous probability distributions defined on the interval [0, 1] and parameterized by two positive shape parameters. It is particularly useful in modeling probabilities such as the chance of heads in coin flips. The shape of the distribution changes with different parameter values, which in this context represent the production characteristics of the coin factories.

Quantile Function and Critical Region

The quantile function, or inverse cumulative distribution function, is used to determine the value below which a given percentage of the data falls. In hypothesis testing, this function helps to construct the critical region. For this problem, the beta_inv function is employed to find the critical threshold such that there is a 5% probability that a coin from factory A yields an observed probability of heads less than this threshold.

Power of the Test

The power of a test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. Here, it measures how effectively the test distinguishes coins from factory B from those of factory A. Computing the power involves evaluating the probability that a coin from factory B falls in the rejection region determined under the null hypothesis about factory A.

Transcript

00:01 So we are assuming that the proportion of heads is 0 .5, and alternately it's not.

00:08 So the two -tail test, and the number of tosses is going to be 36, and we're talking about the rejection region being if the number of heads minus 18 is greater than or equal to 4.

00:26 And so we think about what that means.

00:29 18 is here, and if we have four higher, which is going to take us up to 22, and there's 22, and four lower is going to take us down to 14.

00:40 So anytime you get more than, more than 22, you would reject, or if you get less than a 14, you reject.

00:49 But we're doing a proportion test.

00:51 So these are out of 36, out of 36, out of 36.

00:55 So if our proportion is higher than 22 out of 36, that would cause us to reject or lower than 14 out of 36.

01:04 And again, this is equivalent to, what, 7 out of 18, and this is equivalent to 11 out of 18.

01:10 And this we know is 0 .5 or 1⁄2.

01:13 So we know that we take the, if we look at those intervals, that the margin of error from here is going to be the z value times the square root.

01:26 Of the proportion times one minus the proportion over n.

01:31 And we know that that margin of error is this difference.

01:35 So the, let's say, 436.

01:39 This difference is 4 out of 36.

01:42 So the margin of air is 4 out of 36, which is 1 9th.

01:47 And we know that that equals the z value times the square root of 1 1 1⁄2 times 1 1 1 1 1 1 half over 36.

01:57 And when we do the algebra on this, this becomes one ninth is equal to z times, and we can square root this and get half times one six, which is one 12.

02:08 So multiplying both sides by 12, we find out that the z value here is three goes into there and three goes into there four thirds.

02:19 So it's about 1 .33 or 1 .33 repeating.

02:24 And then we want to find what this area is.

02:27 Above and double it so that's going to be what we would consider in our textbook what the alpha level was so when i did my normal cdf or you look this value up in the table and then double it i get that that comes out to be 0 .0912 and then times 2 so approximately 18%.

02:53 Now if your textbook is saying that this one level over here is your alpha, then you'd only say nine, but we would use 18.

03:02 Now, the next question asks, if the actual value, and this is part b, you basically want to find what is the likelihood of a beta error if the actual proportion is 0 .7.

03:16 So we want to find the probability of not rejecting, so not rejecting, if the actual proportion is equal to 0 .7.

03:30 So what does that look like? that means we're going to get some type of our not rejecting is in between here and here...

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