A wine producer claims that the proportion of its customers...

A wine producer claims that the proportion of its customers who cannot distinguish its product from frozen grape juice is, at most, $0.09$. The producer decides to test this null hypothesis against the alternative that the true proportion is more than $0.09$. The decision rule adopted is to reject the null hypothesis if the sample proportion of people who cannot distinguish between these two flavors exceeds $0.14$. a. If a random sample of 100 customers is chosen, what is the probability of a Type I error, using this decision rule? b. If a random sample of 400 customers is selected, what is the probability of a Type I error, using this decision rule? Explain, in words and graphically, why your answer differs from that in part a. c. Suppose that the true proportion of customers who cannot distinguish between these flavors is $0.20$. If a random sample of 100 customers is selected, what is the probability of a Type II error? d. Suppose that, instead of the given decision rule, it is decided to reject the null hypothesis if the sample proportion of customers who cannot distinguish between the two flavors exceeds $0.16$. A random sample of 100 customers is selected. i. Without doing the calculations, state whether the probability of a Type I error will be higher than, lower than, or the same as in part $\mathrm{a}$. ii. If the true proportion is $0.20$, will the probability of a Type II error be higher than, lower than, or the same as in part c?

A wine producer claims that the proportion of its customers who cannot distinguish its product from frozen grape juice is, at most, $0.09$. The producer decides to test this null hypothesis against the alternative that the true proportion is more than $0.09$. The decision rule adopted is to reject the null hypothesis if the sample proportion of people who cannot distinguish between these two flavors exceeds $0.14$.
a. If a random sample of 100 customers is chosen, what is the probability of a Type I error, using this decision rule?
b. If a random sample of 400 customers is selected, what is the probability of a Type I error, using this decision rule? Explain, in words and graphically, why your answer differs from that in part a.
c. Suppose that the true proportion of customers who cannot distinguish between these flavors is $0.20$. If a random sample of 100 customers is selected, what is the probability of a Type II error?
d. Suppose that, instead of the given decision rule, it is decided to reject the null hypothesis if the sample proportion of customers who cannot distinguish between the two flavors exceeds $0.16$. A random sample of 100 customers is selected.
i. Without doing the calculations, state whether the probability of a Type I error will be higher than, lower than, or the same as in part $\mathrm{a}$.
ii. If the true proportion is $0.20$, will the probability of a Type II error be higher than, lower than, or the same as in part c?

Key Concepts

Sample Size and Variability

Sample size plays a crucial role in hypothesis testing because it affects the variability (or standard error) of the sample estimate. Larger samples tend to produce more precise estimates by reducing the standard error, which affects the probabilities of Type I and Type II errors. A change in sample size will shift the sampling distribution and can make the statistical decision more or less conservative.

Decision Rule

In hypothesis testing, the decision rule defines the criterion for rejecting the null hypothesis based on the sample data. This rule is determined by setting a threshold or critical value for the test statistic (for example, a specific sample proportion), such that if the observed value exceeds (or falls below) the threshold, the null hypothesis is rejected.

Power of a Test

The power of a test, defined as 1 minus the probability of a Type II error (1 - ?), measures the ability of a test to correctly reject the null hypothesis when the alternative hypothesis is true. Increasing the sample size, or adjusting the decision threshold, can increase the power of the test, thus reducing the chance of missing a true effect.

Type II Error

A Type II error happens when the test fails to reject the null hypothesis even though the alternative hypothesis is true. This error is represented by the probability ?, and it affects the test’s sensitivity. A high probability of a Type II error means the test is less capable of detecting an effect when one exists.

Null and Alternative Hypotheses

This concept involves setting up a null hypothesis (H0), which is a statement assumed to be true for the purposes of testing, and an alternative hypothesis (Ha), which is the claim that the test aims to provide evidence for. In hypothesis testing, the null hypothesis typically represents the status quo or a claim of no effect, while the alternative represents the claim of interest (such as a higher proportion or a different mean).

Type I Error

A Type I error occurs when the null hypothesis is rejected even though it is actually true. It represents a false positive result in hypothesis testing. The probability of committing a Type I error is denoted by the significance level (?) and is controlled by the decision rule set for the test, which defines the critical region where the null hypothesis is rejected.

Transcript

00:02 So we'll be assuming that the proportion is no more than 9 % or 0 .09, and alternately, that the proportion is more than 0 .09.

00:11 And we're told that the rejection region that they're going to use is if they get a p -half that is greater than 0 .14.

00:19 And in part a, it says if our sample size is 100, what would be the probability of a type 1 error? so in other words, here is what we're assuming as the center of our sampling distribution, and this is where we are going to reject.

00:41 So we need to know what that z value is that corresponds with that, and so the z value will end up being 0 .14 minus 0 .09 divided by, and we'll have that 0 .09 times 0 .91, divided by 100.

00:59 And that z value, let me quick do my calculation here.

01:04 So you have 0 .05 in the numerator divided by the square root of .09 times .91 divided by that 100.

01:13 And that z value comes out to be 1 .747, i'll just call it 75.

01:22 And then when we, i have to use my inverse normal actually now to figure out.

01:27 What that value is or actually the normal cdf.

01:32 I don't happen to have my calculator here or my book here with a table.

01:38 So we'll just quick find what that area is.

01:43 And that probability comes out to be about 4%.

01:49 So the area above this with that z value is approximately 4%.

01:54 So our probability of a type 1 error would be about 4%.

01:57 Now on part b, the question changes and now our sample size is 400 and we want to know what the probability of a type 1 error is.

02:08 So we can see that what we need to do is change this to 400 and find what that z value is.

02:16 So let me quick go up and we're going to just change that calculation of that value to 400.

02:24 And when we do, we get a z value that is equivalent to.

02:29 It's very large.

02:30 3 .49.

02:32 And once again, i'm going to have to go back and use my normal cdf to find what that is.

02:39 I don't happen to have a table here.

02:41 I could look it up online.

02:42 I could sure we could do that, couldn't i? in any case, i end up getting that that value is 0 .0024.

02:51 So our likelihood of a type 1 error would now be reduced to that value.

02:58 So we can see that increase in the sample size dramatically decreased if we have a particular level of type 1 error.

03:06 It changes that z value dramatically.

03:10 Now, next part is we want to find on part c, if our sample size is 100, what is the likelihood of a type 2 error? if the actual proportion is 0 .2.

03:27 And our type 2 error is the likelihood that we don't reject, or we quote, accept the null...

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