"MigrainesAccording to the Centers for Disease Control, 15.2% ...

Question

"MigrainesAccording to the Centers for Disease Control, 15.2% of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches.

a. Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist’s techniques.

"

b. A sample of 500 American adults who participated in the massage therapist’s program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist’s program.

"

Accepted Answer

Welcome back everyone. In this problem, a city department reports that historically 30% of households recycle regularly. A new educational campaign claims it increases the recycling rate. A random sample of N equals 600 households after the campaign produces data that leads to rejection of the null hypothesis, which statement best supports the campaign? A says there is sufficient evidence to conclude that the recycling rate remained unchanged. B says there is sufficient evidence to conclude that the campaign increased the recycling rate. C says that there is sufficient evidence to conclude that the campaign decreased the cycling rate. Sorry, the recycling rate and the D says there is insufficient evidence to conclude anything about our proportion P. Now if we're going to figure out which statement best supports the campaign, let's first establish our hypotheses and check our conditions to see if uh the, the conclusions will be valid and we're going to get our standard error as well. Now notice that the city department reports that 30% of households recycles regularly while a new campaign claims it increases the recycling rate. So the alternate hypothesis will be the claim and thus we can let our another hypothesis be that P or proportion equals 0.30. Where our alternate hypothesis is that P is greater than 0.30 because the campaign claims it increases the recycling rate. Now if we check the conditions here, checking the conditions. Hm. Where for. Our sample size and multiplied by our proportion is greater than our equal to 10, and that N multiplied by 1 minus our proportion is also greater than or equal to 10, then we know that our sample size N is 600 and our proportion is 0.3. So, NP is going to be equals to 600 multiplied by 0.3, which equals 180, and that's greater than 10. And N multiplied by 1 minus P will be equal to 600 multiplied by 0.70, which is 420. And that's also greater than or equal to 10. So it satisfies both of these conditions. Next, for a standard error. OK, the standard error. SE Is going to be equal to the square root of p multiplied by 1 minus p divided by N. Now if we substitute those values, that's the square root of 0.3 multiplied by 0.7. OK, divided by 600. And that would be approximately equal to 0.01871. Now, if we use the formula that says Z equals our sample proportion, minus our proportion divided by the standard error, the rejection of formal hypothesis in a right tail test implies the observed Z using, let me write that here, U Z equals. P minus P not divided by SE, OK. Then rejection of or null hypothesis like our problem told us in a right-tailed test. OK, implies. Let me just write out the statement I started earlier, that the observed value of Z was sufficiently positive. OK Sufficiently positive. And The P value Was less than the significance level. So what does that mean for us? Well, in this case, since HO was rejected for our our alternative hypothesis that P is greater than 0.3, we can conclude that there is sufficient evidence that P is greater than 0.3, supporting the campaign's claim. If we go back to our answer choices, if we examine our statements. In statement A, it says there is sufficient evidence to conclude that it remained unchanged, but rejection of the non hypothesis contradicts concluding equality. Therefore, this choice is incorrect. In answer choice B, there is sufficient evidence to conclude that the campaign increased the recycling rate. This is the correct answer. Based on the information that we have. If we review the remaining choices, whereas in C, it says there's evidence to conclude that the campaign decreased the recycling rate, the campaign claimed an increase. So rejection would not necessarily support that P is less than 0.3. Therefore, this choice is incorrect. And in answer choice B, it says that there is insufficient evidence to conclude anything about P, but rejecting our non hypothesis provides evidence for the alternative. So claiming insufficient evidence is incorrect here. That's how we know that answer choice B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Null Hypothesis and Alternative Hypothesis

Hypothesis Testing and Significance

Sample Size and Its Impact on Statistical Power

Watch next