[APPLET] A researcher claims that the mean age of the resident...

Question

[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

Table displaying ages of 30 residents, with values ranging from 21 to 87 years, for hypothesis testing analysis.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A factory manager claims that the average daily output of machines is greater than 200 units. To test this claim, a random sample of 30 machines shows a mean output of 206 units per day. The population standard deviation is known to be 12 units. At the 0.10 level of significance, is there sufficient evidence to support the manager's claim. Awesome. So it appears for this particular prom we're asked to take all the information that is provided to us by the prom itself and noting that we're told that we have a level of significance of 0.10 and we're asked to determine whether or not there is sufficient evidence to support the manager's claim. Which once again, the manager is claiming that the average daily output of machines is greater than 200 units. So trying to determine once and for all if there's sufficient evidence to support the manager's claim based on all the information that is provided to us by the problem itself. So with that in mind, now that we know that we're ultimately trying to determine that we're testing the claim about the proportion mean with a known standard deviation. And because we know once again that we're testing a claim about a population mean with a with a known standard deviation, because of this information we know as a result, as we should recall, that we can use a Z test for the mean. So with that in mind, our first step is we need to state our hypothesis for this particular prompt, and let us denotem to be the daily mean output. So this is the true mean daily output will be new. So once again, letting mu be the true mean daily output, we could state that the null hypothesis H subscript 0 H0 will state that mu is equal to 200, and the alternative hypothesis HA will state that mu is greater than 200. Fantastic. And let us note that as we should recall, we need to be able to recognize that based on the information that we just discovered that this will be a right tail test which we'll call RT for short. So this is a right tail test, so moving right along here. Our next step, our second step is we need to identify our known variables. We're told that X bar, the sample mean, is equal to 206. We're also told that m 0 is equal to 200, and this is our hypothesized mean. We're also told that sigma, the population standard deviation. is equal to 12. Our sample size N is equal to 30, and finally, our level of significance alpha is equal to 0.10. Fantastic. So our third step now is we need to recall and use the Z test formula. So we need to use the Z test statistic because we are told the value of sigma, the population standard deviation. That means we can recall and use Z equals X minus m 0 divided by sigma divided by the square root of N. And we need to recall a note that this formula measures how far the sample mean is from the hypothesized mean in terms of standard errors. So, with that in mind, we can now plug in our known variables and solve for Z, so this will be equal to 206 minus 200. Divided by 12 divided by the square root of 30, which when we plug in this expression into our calculator and we round to two decimal places, we will find that it's approximately equal to 2.74. Awesome. So, Moving right along here. Our 4th step. So that was our 3rd step, and I guess technically you could say your 4th step was when we plugged in our known variables. So that will mean that our 5th step is to find the critical Z values. So once again, since we are noting that we're dealing with a right tail test and we're told that we have a significance level of 0.10, that will mean that when we are trying to determine the critical Z value, that will mean that our critical Z value, which we'll call Z subscript C, will be equal to or rather it will be approximately equal to 1.28. And once again, we know that it's approximately equal to 1.28 because we need to use our standard normal distribution table, which should be table 4 in your elementary statistics textbook that's 8th edition. So With that in mind, Our sixth step is we need to make our final decisions, and since Z is equal to 2.74, which is greater than 1.28, that means we can indeed reject the null hypothesis, so I'll put an X next to H0. So once again, because 2.74 is greater than 1.28, we can reject the null hypothesis. So that means we can make our final conclusions as such. There is sufficient evidence at the 0.10 significance level to support the manager's claim that the average daily output is more than 200 units. That will mean when we go to look at our multiple choice answers, or there is no multiple choice answers. We will determine And our final answer, without a doubt, once again, will be that. There is sufficient evidence to support the claim that the average output is more than 200 units, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Hypothesis Testing

Significance Level (α)

Z-Test for Means

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