A researcher is comparing average number of hours spelt per night by college students who work part-time versus those who don't. From survey data, they calculate x ˉ 1 = 6.82 x̄_1=6.82 hours and x ‾ 2 = 6.57 \overline{x}_2=6.57 hours with a margin of error of 0.41. Should they reject or fail to reject the claim that there is no difference in hours slept between the two groups?

Alright, everybody. So let's get some more practice here working with confidence intervals when we have two samples or two means. Let's take a look at our problem. So a researcher is comparing the average number of hours slept per night by college students, which is probably a sadly low number, for students who work part time versus those who don't. Now from survey data, they calculate a couple of different sample means. The first one is x one bar is equal to six point eighty two hours. The second one is six point fifty seven hours, which is x two bar. Now the margin of error that they calculate for this is 0.41. Again, in all these types of problems here, you're just gonna assume that the numbers they've collected and the calculations that they've done are all correct. Right? So we're gonna assume that all this stuff is correct here. Now what we're supposed to do in this problem is not construct an entire confidence interval because most of the stuff is already given to us, but we're just supposed to figure out whether we should reject or fail to reject a claim that's being made, or in other words, a hypothesis. The hypothesis or the claim is that there is no difference in hours slept between the two groups. Alright? So again, what's the connection between confidence intervals and hypothesis tests? Well, remember, the rule here is that if you have a confidence interval, remember your confidence interval is always gonna be made up of two numbers. Right? Your lower bound is gonna be your point estimator x one bar minus x two bar, and that's gonna be minus e. And then in the upper bound, it's gonna be x one bar minus x two bar plus e. Right? It's always those two numbers. Now the rule here is if your interval does not include zero, then then then you have you have confidence or you have evidence that the true difference in means is actually not zero. So in other words, you would reject the null hypothesis. Alright? So remember the rule for rejecting is if it does not include zero versus the opposite, which is that if it does include zero, then you would fail to reject. So it's, again, a little bit backwards here, but always remember they're kind of opposites. Oops. So if it does include zero, remember, that means that it's plausible that the true difference in means actually might just be zero, might be nothing, it might be no difference. Right? So, basically and if it does not include zero, then, therefore, you can reject it because you're confident that zero is not inside of that interval. Okay? So in other words, we just basically have to figure out using the information that we're given, does zero fall into this interval or not because we have all of these numbers here. We have our x one, our x two, and we have our e, which is our margin of error. So let's go ahead and build out this confidence interval. Alright? So our confidence interval is just going to be 6.82 minus 6.57. That's our point estimator. So our point estimator here, minus our margin of error, which is 0.41, and our upper bound is gonna be same two numbers again, 6.82 minus 6.57. This is our, like, best guess of what the true, difference actually is, and then we're gonna add that point 41. Alright? So, again, we can't we're sort of just trusting that these numbers are accurate, but that's what we're gonna do. Alright? So in other words, our interval ends up being we could actually just plug all these numbers in sort of in this one step. That's what I did. Your interval is going to be we get negative 0.16 on the lower end and then positive 0.66 on the upper bound. Alright? So now does this include zero? I always like to just basically just write this. So include zero question mark. And if it does, then we are going to fail to reject. And if it does not include zero, then we reject. Clearly, we can see here that this interval does, in fact, include zero. Therefore, we are going to fail to reject. Alright? So that is always gonna be the conclusion when your interval includes zero. So you're gonna fail to reject. Therefore, there's gonna be not enough evidence that there is actually a significant difference in average, you know, hours number of hours slept by college students. Alright? Alright. So that's really all there is to it. Remember, always use to remember the rule, construct your confidence interval, and figure out if zero is in that interval or not. That's always gonna tell you whether you reject or fail to reject that claim. That's it for this one, folks. Let me know if you have any questions.

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

Statistics for Business

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

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Multiple Choice

A researcher is comparing average number of hours spelt per night by college students who work part-time versus those who don't. From survey data, they calculate $x ˉ sub 1 equals 6.82$ hours and $x overline sub 2 equals 6.57$ hours with a margin of error of 0.41. Should they reject or fail to reject the claim that there is no difference in hours slept between the two groups?

Reject

Fail to reject

There is not enough information to answer the question

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis states that there is no difference in the average number of hours slept between the two groups (H₀: μ₁ = μ₂). The alternative hypothesis states that there is a difference (Hₐ: μ₁ ≠ μ₂).

Step 2: Identify the sample means (x̄₁ = 6.82 and x̄₂ = 6.57) and the margin of error (ME = 0.41). The margin of error provides the range within which the true difference in means is likely to fall.

Step 3: Calculate the confidence interval for the difference in means. The difference in sample means is Δx̄ = x̄₁ - x̄₂ = 6.82 - 6.57. The confidence interval is given by Δx̄ ± ME, which means you will calculate the lower bound as (Δx̄ - ME) and the upper bound as (Δx̄ + ME).

Step 4: Check if the confidence interval includes 0. If 0 is within the confidence interval, it means there is no statistically significant difference between the two groups, and you fail to reject the null hypothesis. If 0 is not within the confidence interval, you reject the null hypothesis.

Step 5: Based on the confidence interval analysis, determine whether to reject or fail to reject the null hypothesis. Since the problem states 'Fail to reject,' it implies that the confidence interval likely includes 0, meaning there is not enough evidence to conclude a difference in hours slept between the two groups.

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