Test the claim about the population mean μ μ at the given level of significance. Assume the population is normally distributed. Find the P P -value and determine whether you should reject or fail to reject the null hypothesis. Claim: μ > 52 \mu>52 , α = 0.10 \alpha=0.10 Sample: x ˉ = 53.1 x̄=53.1 , s = 4.7 , s=4.7, n = 20 n=20

Everybody. So now that we understand the basic difference between a hypothesis test, where you know sigma versus where you don't, hopefully you had a chance to work out this practice problem here. Let's go ahead and get started here and see how to work it out together. So we're gonna be testing a claim about this population mean, which is at the given level of significance. So we've got alpha is equal to 0.1. Now we're assuming the population is normally distributed, which takes care of a couple of criteria for us, and we're going to be finding, as we've done before in the past, we're going to be finding the p value and determine whether you should reject or fail to reject this null hypothesis. We don't have a story here. It's similar to other problems where we've just given a bunch of numbers, but let's just go ahead and stick to the steps here. The first thing we have to do is we have to set up our ho and r j, the null and alternative hypothesis. Alright? So we've got ho and How do we write this with you using symbols? The claim that's being made over here is that mu, the mean, is greater than some number 52, whatever that represents, but that's the claim that's being made. Now, the null hypothesis is always that mu is equal to some number here. So basically, if the claim is mu is greater than 52, then the working assumption or the default assumption that we make is that mu is equal to 52, whereas the claim that mu is greater than 52 ends up being the alternative hypothesis. Usually, in a claim, if this symbol here is not an equal sign, that's actually and not h o. Alright? So this is how you can actually write both of those statements from this single one over here. Now, by the way, because we're using a greater than sign over here, this actually means this is going to be a right tailed test. Alright? So now now that we have our HO and squared away, now we're going to go ahead and calculate our test statistic. Now remember, because we don't have the population standard deviation here, if you look at all of these numbers, we only have x bar, s, and n. We don't have sigma. Now, that means that we're not going to be using a z score over here. We're going to be using a t score, from the t distribution. Now remember, if the equation for this ends up being very similar to the z score equation, we have x bar minus mu in the numerator, but instead of sigma, we have s in the denominator divided by the square root of n. Alright, so just be really careful here about those equations. They can be easy to sort of confuse, but But typically, what's going to happen is if you have a bunch of numbers here because you have s, that's the number that you have to plug into this anyways. Alright. So we're going to have the sample mean, which is 53.1 minus the mu, which we're just using as 52 for now. That's what our assumption is. Divided by we've got 4.7 divided by and we've got the square root of 20. Now if you go ahead and work this out what you're going to find here is that this t score is equal to 1.05 if you round to the nearest hundredth place. Alright? Now, again, what this right tailed test means and the fact that we got a positive t score over here is it's kind of like if you had a sort of normal or or sort of a t distribution that kind of looks like this, where our zero point is over here. This is our mu. This is our mu equals 52. What we actually ended up getting over here was we got a number that's sort of right about over here. So you got t is equal to 1.04. Now, what we're basically trying to do over here in a right tailed test is trying to figure out what's the area that's to the right of this t score over here. How do we do this? We're going to take this t score, and we're going to turn it into a probability. So what is the p value over here? What we're going to basically just do, what is the probability that t is greater than this number that I just found over here, 1.05? Now, you can use a t table for this, or you can use a graphing calculator for this. Either way, just knowing the t score isn't enough. We also have to figure out what is the degrees of freedom. So in this score over here, 1.05, what's the degrees of freedom? What we have at that n is equal to 20 over here, which means that our degrees of freedom is going to be one subtracted from that number. So, it's going to be 20 minuteus one or the degrees of freedom of 19. So, we have to use the t score, which is 1.05, and the degrees of freedom of 19, and you're going to be looking at to the right of that. Again, use a graphing calculator or a t t table. Go ahead and pause the video so you see we can work out this and make sure you get the same number that I do. But what you should get by this p value is that it's equal to 0.154. Alright? That's going to be our p value, which is basically just the area that is shaded to the right of that number. Okay. Now as for our conclusion, works the same way as any other hypothesis test. We're going to take this p value, we're going to compare it to our alpha, and we're going to determine whether to reject or fail to reject that null hypothesis. Now, our p value is 0.154 and our alpha is 0.1. These numbers are pretty close to each other. However, what happens is the p value is going to be greater than that alpha Because it's greater, we're going to go ahead and fail to reject. Right? That's going to be our answer over here. So that's our p value, which is this number over here. And because of that, we're going to fail to reject, and that's our conclusion. Alright, folks. That's for this one. Let me know if you have any questions. Thanks for watching.

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Means

Statistics for Business

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Means

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Multiple Choice

Test the claim about the population mean $μ$ at the given level of significance. Assume the population is normally distributed. Find the $P$ -value and determine whether you should reject or fail to reject the null hypothesis.
Claim: $mu is greater than 52$ , $alpha equals 0.10$
Sample: $x ˉ equals 53.1$ , $s equals 4.7 comma$ $n equals 20$

P-val = 0.154; Reject the null hypothesis.

P-val = 0.154; Fail to reject the null hypothesis.

P-val = 0.308; Reject the null hypothesis.

P-val = 0.308; Fail to reject the null hypothesis.

Verified step by step guidance

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The claim is μ > 52, so the null hypothesis is H₀: μ ≤ 52, and the alternative hypothesis is H₁: μ > 52. This is a one-tailed test.

Step 2: Determine the test statistic formula. Since the population standard deviation is not provided and the sample size is small (n = 20), use the t-test formula: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Step 3: Calculate the degrees of freedom (df) for the t-distribution. The degrees of freedom are given by df = n - 1. In this case, df = 20 - 1 = 19.

Step 4: Use the calculated t-value and the degrees of freedom to find the P-value. The P-value represents the probability of observing a test statistic as extreme as, or more extreme than, the calculated t-value under the null hypothesis. Since this is a one-tailed test, the P-value corresponds to the area to the right of the calculated t-value in the t-distribution.

Step 5: Compare the P-value to the significance level (α = 0.10). If the P-value is less than α, reject the null hypothesis. If the P-value is greater than or equal to α, fail to reject the null hypothesis. Based on the given P-value, determine the correct conclusion.

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