Test the claim about the population mean μ \mu 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 ≠ 1020 μ≠1020 , α = 0.01 α=0.01 , σ = 85 σ=85 Sample: x ˉ = 990 x̄=990 , n = 40 n=40

Everybody. So let's take a look at this practice problem. I hope you got a chance to pause it and see if you work it out on your own now that we've done a couple of examples together. So we're gonna be testing a claim about some population mean, mu, at some given level of significance. That's alpha, which is equal to 0.01. Now, we're going to assume the population is normally distributed. This helps take care of a couple of criteria that we have. We're going to be finding the p value. Remember, that's typically what we do sort of near the end of the hypothesis test and determine whether we should reject or fail to reject this HO. All right? So let's take a look here. Again, this is like others problems we've sort of seen so far where there's not really a story that you have to interpret. It's just a bunch of numbers you kinda have to plug and chug. But, nonetheless, we're gonna just go ahead and stick to the steps. Now we're gonna have a bunch of a bunch of information as to the claim and sample, but first, what we're gonna do is we're gonna set up our initial hypotheses. This is our HO and our Now what we're told here is that some claim is being made that the population mean is not equal to some number, which is 10 a 20, but that's not necessarily our null hypothesis. Remember, our null hypothesis is always mu equals some number. If the claim is that it's not ten twenty, then your default assumption is that it is ten twenty. So your HO is going to be mu equals ten twenty. And what's our alternative hypothesis? What's the sort of symbol that we're using about this population mean? Well, it's basically just what's being made in the claim over here that mu is not less than, it's not greater than, it's just not equal to that number, again, ten twenty. Usually what happens here is in a claim that's being made, if it doesn't contain an equal sign, that's gonna be the the alternative hypothesis, actually, not the null hypothesis. Okay? So this is our h o and a which, by the way, this means that we're working with a two tailed test. This can be really important in here when you calculate our p value. Okay? So this is our h o and r j. So now what we're going to do is we're going to go ahead and calculate our test statistic. Again, we're going to use a z score over here because we actually know what the population standard deviation is. It's 85. Right? So in our z score, what we're going to do is, if you remember, we're going to take our x bar minus mu, and we're going to divide it by our sigma divided by the square root of m. Plugging those numbers in, what we get is our sample mean, which is 990 minus a 20 divided by the sigma, which is 85, and then we have divided by the square root of the sample size, which is 40. If you go ahead and plug in all those numbers here, what you should get for your z score is you should get negative 2.23. Alright? So this is basically just sort of a z score that tells us how far away from the average, assuming that it is ten twenty, this sample would actually be. How do we turn this into sort of a probability? Well, we're going to convert this into a p value. So our p value over here, how do we figure this out? Because we are doing a two tailed test, right, because we just found that this is going to be a two tailed test because of our alternative hypothesis. If you were to sort of graph this out like this, what that actually means is that we actually have something like this going on, which is we have a negative 2.23 over here. This is our z score. This is our 2.23. This would be something like zero. We're actually doing is we're finding the area that is off to both of those tails over here. So we have to do is where our p value is going to be two times the probability that z is going to be to the left or right of this number over here. So the z is to the left of negative 2.23. Basically, you're just going to take the z score and the probability, and you're just going to double it. Right? So you're just going to have to multiply it by two. So if you actually go ahead and work this out here, again, use a z table or use a graphing calculator, that can be really helpful with this. What you're actually going to find here is you're going to find a p value of 0.026. If you got something like 0.013, then it's because, again, your z table is only taking into consideration only half of that probability. You have to multiply it by two. Okay? So if you got 0.013, it's because you forgot to multiply it by two. Be very careful. It's a very easy mistake to make. Okay? So this is gonna be 0.026. Now we're gonna do for our last step over here. We're gonna have to take this probability, and we're going to compare it to our alpha in order to reject or fail to reject. So, we have p value and alpha, and I'm just going to write those numbers over here. So, we have 0.026 and our alpha which is 0.01. And clearly, we can see here that even though these numbers are actually really, really close to each other, the p value is actually be slightly greater than your alpha. Alright? So and, therefore, when it comes to reject or fail to reject, what is it gonna be? So this is gonna be our reject, and this is going to be our fail to reject. Usually, I just abbreviate that as FDR. So which one is going to be? Remember, when p value is greater than alpha, you're always going to fail to reject. Right? So there you go. You're going to fail to reject this null hypothesis. So in other words, there's really just not enough evidence to say that the mu, whatever this is, is actually not equal to ten twenty. Alright? So that's really all there's to it, folks. Hopefully, you got that one right. Let me know if you have any questions, and thanks for watching.

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Means

Statistics

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: $mu is not equal to 1020$ , $alpha equals 0.01$ , $sigma equals 85$
Sample: $x ˉ equals 990$ , $n equals 40$

P-val: 0.026; Reject the null hypothesis

P-val: 0.026; Fail to reject the null hypothesis

P-val: 0.013; Reject the null hypothesis

P-val: 0.013; Fail to reject the null hypothesis

Verified step by step guidance

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 1020, and the alternative hypothesis is H₁: μ ≠ 1020. This is a two-tailed test because the claim is that the population mean is not equal to 1020.

Step 2: Calculate the test statistic using the formula for the z-test: z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size. Substitute the given values: x̄ = 990, μ = 1020, σ = 85, and n = 40.

Step 3: Determine the P-value for the calculated z-test statistic. Since this is a two-tailed test, the P-value is found by doubling the area in the tails of the standard normal distribution corresponding to the absolute value of the z-test statistic.

Step 4: Compare the P-value to the significance level α = 0.01. If the P-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Step 5: Based on the comparison, make a decision about the null hypothesis and interpret the result in the context of the problem. If the null hypothesis is rejected, it means there is sufficient evidence to support the claim that the population mean is not equal to 1020. If the null hypothesis is not rejected, it means there is insufficient evidence to support the claim.

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