In Exercises 29–32, test the claim about the population mean a...

Question

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≤ 22,500; α = 0.01; α = 1200

Sample statistics: x_bar = 23,500, n = 45

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a random sample of 38 observations has a sample mean of 19,820. Assume that the population is normally distributed with a known standard deviation of sigma equal to 950. At the 0.01 level of significance, test the claim that the population meanmu is less than or equal to 19,000. So we're going to begin by writing down our hypotheses. Now, for our null hypothesis, each knot, we're gonna have our claim that mu is less than or equal to 19,000. And the reason why we use this as our null hypothesis is because this statement has an equality in it, since we have since we have less than or equal to in this statement. That's why it's our null hypothesis. Now, for the alternative hypothesis, HA, We're going to have the complement of this, and the complement of our no hypothesis is going to be mu is greater than 19,000. And notice that this statement also has a strict inequality since we have just greater than. And since we have a greater than sign in this statement, that means that our hypothesis test is going to be right tailed. Now, we also have some other information given in this problem. We're given our sample size in, which is going to be equal to 38, since we have 38 observations. We're also given our sample mean, X bar, and that is going to be equal to the 19,000. 820 And we're also given our population standard deviation, sigma, which is equal to 950. And finally, we're given our level of significance, which is 0.01. And so that means that alpha is going to be equal to 0.01. Now, since N, which is equal to 38, is greater than 30, and the population standard deviation is known, as well as the sample being random, that means that the appropriate test in this case is going to be the Z test. So, we'll need to calculate our test statistic for a population mean with a known population standard deviation. So for that, recall your formula 4 Z. So Z is going to be equal to the quantity of X minus mu in quantity divided by the quantity of sigma divided by the square root of N in quantity. And we actually have values for all those quantities, so we'll substitute in those values. So Z is going to be equal to. The quantity of 19,820. Minus 19,000. In quantity, divided by the quantity of 950. Divided by the square root of 38 in quantity. And when we evaluate this expression, this is gonna be approximately equal to. 5.32. Next, we need to look up our critical value. And in our case here, we're gonna be looking at the critical value or a right tail test with alpha equal to 0.01. And so looking at that value, we see that our critical value, Z sub C is going to be equal to 2.33. So, now we just need to compare our critical value here to our calculated Z value. And we see that our calculated Z value, is actually greater than our critical value Z C. Since 5.32 is greater than 2.33. And this puts our calculated Z-value in the rejection region. Since the rejection region for the right tail test is going to be the region that is greater than our critical Z-value. So, that means that we're actually going to reject our null hypothesis. So we're going to reject each knot. Which means that there is sufficient evidence. To reject our claim. So there is sufficient evidence to reject the claim that the population being new is less than or equal to 19,000. So that is going to be the answer for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≤ 22,500; α = 0.01; α = 1200
Sample statistics: x_bar = 23,500, n = 45

Key Concepts

Hypothesis Testing

Level of Significance (α)

Sample Mean and Standard Error

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