Performing a Sign Test In Exercises 7–22, (b) find the critica...

Question

Performing a Sign Test In Exercises 7–22, (b) find the critical value,

Social Media A research group claims that the median age of the users of a social media website is greater than 30 years old. In a random sample of 24 users, 11 are less than 30 years old, 10 are more than 30 years old, and 3 are 30 years old. At , can you support the research group’s claim? (Adapted from Pew Research Center)

Accepted Answer

Hi everybody. Welcome back. The next question says, a tech reviewer claims that the median battery life of a certain brand of smartwatch is greater than 20 hours. Out of 18 tested units ignoring ties, 5 had more than 20 hours and 13 had less. Conduct a right-tailed sign test at the 0.05 level of significance. What is the critical value? A X is less than or greater than or equal to 18. B X is greater than or equal to 10. C. X is greater than or equal to 13, or D X is greater than or equal to 15. So first, let's look to define our hypotheses. We're talking about a median value. So, our null hypothesis will then be that the median Is equal to 2 hours. While our alternative hypothesis About the median Says that our tech reviews claiming that the median is greater than 20. The median is greater than 20 hours. So our samples, it's at 18 units were tested, and that ignores the ties. So our N is going to equal 18. So we need to use the binomial distribution. We can't use a normal distribution to approximate this. So we have our N equals 18. Since we're looking at a median, our probability. Of success, so in this case being greater than. The median, but more correctly, we're looking at our null hypothesis. So that just says the median is equal to 20 hours. So our probabilities of a of a success, we just pick which a sign test, which is positive, and which is negative. So we'll say positive equals. Greater than 20. And negative is less than 20. So our probability, looking at the greater than 20 results, says that we had 5 of those and 13 had less. So, the probability of our positive event is going to equal 0.5. Since the null hypothesis with a median value, we have an equal possibility of getting a result positive or negative. We are interested in those positive results since what we're testing, we're looking for. Watches with greater than 20 hours of battery life. So what we're looking for is the smallest. Value of X. Such that the probability Of X being greater than or equal to. That X is less than or equal to 0.05. So again, that tells us assuming that our median is 20 hours, our null hypothesis is true, the probability of our Specific number of possible samples being greater than are equal to that critical value X is needs to be less than or equal to our 0.05, which is our. Alpha value So, we use a binomial distribution table or a calculator and try some bodies of X. Looking for the p value associated with them, so let's try with. X being greater than or equal to 13, so our small x being 13. And the P value that corresponds with that, so with 13 samples being positive, or 13 being positive and a size 18 sample with this probability of 0.5, is approximately 0.048. So that's pretty close to our 0.05 value, but it's still less than alpha. So, 13's looking like a pretty good candidate, but again, we're looking for the smallest possible value of X. So, we will check on X, the probability of X being greater than or equal to 12. And if we look up that P value, For X being 12, we get approximately 0.091. So that's going to be greater than alpha, so that means that. That probability is too high. So the X critical value that we're looking for is 13. Meaning that if our number of positive signs or values over 20 hours is greater than or equal to 13, then we reject the null hypothesis. So our answer is choice C. That X must be greater than or equal to 13 as our critical value in this case. See you in the next video.

Key Concepts

Sign Test

Critical Value

Hypothesis Testing

Watch next