Randomization: Testing a Claim About a Mean
In Exercises...

Question

Randomization: Testing a Claim About a Mean

In Exercises 9–12, use the randomization procedure for the indicated exercise.

Section 8-3, Exercise 23 “Cell Phone Radiation”

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a tech company claims that the mean battery life of its new smartwatch is at least 24 hours. A skeptical reviewer believes the battery life is less than 24 hours. A random sample of 10 smartwatches resulted in a mean battery life of 24 hours. After simulating 2000 randomizations under the null hypothesis, 7D simulated means are as extreme or more extreme than 24 hours. Use a randomization procedure to test the reviewer's claim at the 0.05 significance level. Now, the first thing that we want to do is write down our hypotheses. So, our null hypothesis, Hno, is going to be mu, which is going to be our mean battery life, is going to be greater than or equal to 24, and our alternative hypothesis, H1, is going to be that mu is less than 24. Now, we're told in the problem that we need to use a randomization procedure to test the reviewer's claim. So, when we cannot assume the data is perfectly normal, especially in this case, when we have small sample sizes, which we had 10 smartwatches as our sample, so instead of using a formula-based method like AT test, randomization simulates the sampling distribution of the sample mean under the assumption that the null hypothesis, true. So how is randomization done here? So here we're going to assume the true meaning of the battery life is exactly 24 hours, which would make our null hypothesis true. We shuffle or simulate many samples, in this case, 2000 random samples based upon this assumption. For each simulated sample, we calculate the sample mean. We then count how many stimulated sample means are as large or larger than the observed sample mean. This gives an empirical estimate of the P-value, the probability of getting a result as extreme as the one that we actually observed, if the null hypothesis is true. Thus, randomization builds a sampling distribution without needing normality or a theoretical formula. So next, we want to calculate our P value based on the information and the problem. So, our P value, based on the information in the problem, is going to be equal to the 7070 simulated means that are as extreme or more extreme than 24 hours. Divided by the total number of randomizations, which is 2000 in this case. And evaluating this fraction here, this is going to be equal to 0.035. So that means there is a 3.5% chance of obtaining a mean of 24 hours, or greater if the true mean battery life was actually at least 24 hours. So we now need to use this p-value to make a decision. And here we're looking at a 0.05 significance level. So that means alpha is going to be equal to 0.05. And here we see that our P value, is actually less than alpha. Since 0.035 is less than 0.05. So that means that we will reject our null hypothesis H knot. And what that means, Is that there is sufficient evidence at the 0.05 significance level to support the reviewer's claim that the mean battery life is less than 24 hours. And so that's gonna be our answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Randomization: Testing a Claim About a Mean
In Exercises 9–12, use the randomization procedure for the indicated exercise.
Section 8-3, Exercise 23 “Cell Phone Radiation”

Key Concepts

Randomization

Mean

Hypothesis Testing

Watch next