Age at First Marriage A marriage counselor claims that the med...

Question

Age at First Marriage A marriage counselor claims that the median age of men at the time of their first marriage is greater than 28 years old. In a random sample of 56 men, 33 were less than 28 years old and 23 were more than 28 years old at the time of their first marriage. At α = 0.05, can you support the counselor’s claim? (Adapted from U.S. Census Bureau)

Accepted Answer

Hello everyone. Let's take a look at this next problem. A health analyst claims that the median age for men to retire is greater than 61 years. In a random sample of 36 men, 14 retired before 61 and 22 retired after 61. At alpha equals 0.05, does the data support the analyst's claim? Yes, there is sufficient evidence to support the claim. The median is less than 61 years. The sample is not representative. Or no, there is insufficient evidence to support the claim. So, let's think about how to set up our hypotheses. This one's a little bit tricky, because our question note is asking, does the data support the claim? So it's not a reject or fail to reject. So if you want data to support the analyst's claim, then the claim should be expressed as the alternative hypothesis. Because then, if you have enough evidence to reject the null hypothesis, that, you know, leads to the idea that you have a sufficient evidence to support the analyst claim, or your data will support it. So, if our alternative hypothesis is the claim, that would say that the median age of retirement is greater than 61. So recall that we're making a statement about the median. We're stating that given a median value, we have half of probability. Or a 50% chance of obtaining a result below the median, and a 50% chance of obtaining a result above the median. So we could say that for our null hypothesis. P, recall that P is the probability of success, and we're going to call success. An age above the medium or positive sign. So if P is our probability of success, then our normal hypothesis is that P equals 0.5, 0.5% chance of getting a result above the median. And then if we're expressing in terms of probability, if the alternative hypothesis is medium is median is greater than 61, that would mean that there's a less than, so P is less than 0.5. That the median. is greater than 61. So, now we need to look at our sample size to think about what kind of test statistic we'll be using. So, we first we'll look at N. Our sample size. So we're going to eliminate any ties, but we don't have any. We're just told that we have 14 retiring before 61 and 22 after. Son would equal 14 + 22. equals a sample size of 36, and it's easy to tell recall the test is that does NP. Is that greater than or equal to 5? Since N is 36, this is definitely larger than 5. So, in that case, we're going to now calculate the mean and standard deviations. We recall that the mean, or mu is equal to NP. So that's going to be our N is 36 in the sample. And that will be multiplied by P. And that is 0.5. So that will equal 18 for the mean. And then we'll calculate standard deviation. Which is that sigma is equal to the square root. Of N multiplied by P, multiplied by 1 minus P or NPQ. That will equal the square root of, then we've got 36 for N. And then 0.5. For each of those probabilities. So we calculate that out. We end up with a value of 3 for our sigma. So now we use a calculating Z score. But we're going to need a continuity correction. Since we have a large enough population, we're going to be approximating with a normal distribution, so we have to have that continuity correction. So, since we're looking at X being greater than, with no equal sign, then we're going to add 0.52. And then the last thing we need to address is Which value for X will we be using? So for that we look back at our. Numbers of Results that were above and results that were both plus and minus, and we choose the smallest. So for X, we're going to choose. The smallest of the two results. So, out of 14 and 22, our smallest number is 14, so X will equal 14. So now we calculate that Z score, remembering that it will equal. Are corrected X value. Minus me. And that divided by our value of sigma. So Z will then equal our corrected X will be our. X value Which is 14, we're going to add 0.5. And then subtract our value of mu, which is 18. How I did keep track of that. And then divide. By our value of sigma for which we have 3. And when I calculate that, I get a Z value equal to. Or approximately equal to. -1.1667. So we can use our Z table and from there get our P value. And recall that we have a. Negative Z value, so we'll be looking at the probability. That Our Z value is less than -1.17. And we get that's approximately 0.12. 10. I'm just going to scoot up a screen just a little bit there to make more space. And so now we're going to compare RP value to our alpha value. So that P value of 0.1210. I Greater than 0.05, which is our alpha value. And since that is the case, we fail to reject. Are no hypothesis. And if you recall our no hypothesis was that the median. was 61, and that we said that P equaled 0.5. So we cannot support the Claim that the median is greater than 61. So our answer will be choice D, no, there is insufficient evidence to support the claim. So we can know that we can eliminate choice A, since that is not the case. Choice B, the median is less than 61 years, that's not what this tells us. This just tells us we can't, we don't have sufficient evidence to reject. The null hypothesis of 61 being the median, and to see the sample is not representative when we have a random sample here. No reason to believe the sample is not representative. So once again, we've got the claim that the median age for men to retire is greater than 61. And we had our sample and gave us numbers of people that retired above 61 and below 61, and alpha of 0.5, and we said, does the data support that analysts claim? And We set the claim of the median being less than 61 as our alternate hypothesis, our null hypothesis being the media was 61 or median, excuse me. And because we ended up with our probability being larger than our significance level, we failed to reject the null hypothesis, and therefore choice D, there is insufficient evidence to support the claim. See you in the next video.

Key Concepts

Median

Hypothesis Testing

Significance Level (α)

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