In Exercises 9 and 10, (a) identify the claim and state Ho and...

Question

In Exercises 9 and 10, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

A researcher claims that the mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. The mean sodium content of 22 randomly selected sandwiches at Restaurant A is 670 milligrams. Assume the population standard deviation is 20 milligrams. The mean sodium content of 28 randomly selected sandwiches at Restaurant B is 690 milligrams. Assume the population standard deviation is 30 milligrams. At α=0.05, is there enough evidence to support the claim?

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 health researcher claims that the distribution of gem attendance across the week is as follows. Sunday, 10%, Monday, 18%, Tuesday, 16%, Wednesday, 14%, Thursday, 12%, Friday, 10%, Saturday, 20%. You randomly survey 50 gym members and ask which day of the week they most often go to the gym. The results are Day frequency. Sunday, 55, Monday, 80, Tuesday, 76, Wednesday, 64, Thursday, 58, Friday, 40, Saturday 127. At alpha equals 0.05, test the researcher's claim. Awesome. So it appears for this particular prompt we're asked to test this researcher's claim at a level of significance of 0.05. That's what alpha denotes is the level of significance. So once again, we're ultimately trying to test this researcher's claim. And the claim once again is, is that the distribution of gym attendances across the week is as follows based on their table that they gave us. So now, now that we know that we're ultimately trying to test this researcher's claim at the specific level of significance value, let us read off our multiple choice answers to see what our final answer might be. So, A is there is sufficient evidence to conclude the distribution is different from what the researcher claimed. There is insufficient evidence to conclude the distribution is different, the claim is reasonable. See if the sample size is too small to conclude the test. And D is the test cannot be completed without knowing standard deviations. Awesome. So with that in mind, our first step that we need to take in order to solve this particular problem is we need to determine our hypotheses. So we need to determine the null hypothesis and the alternative hypothesis. So as we should recall, the null hypothesis is denoted as H subscript zero, which I'll just call it H0 for short. And the null hypothesis, 80 represents the distribution of gym attendance matches the claimed distribution. And for our alternative hypothesis, HA, the distribution does not match the claimed distribution. So, With that in mind, our second step now is we need to calculate the expected frequencies. So let us know that our total responses, which we'll call T subscript R for short, so our total responses. is equal to 500, and since we know that our total responses are equal to 500 and we're asked to calculate the expected frequencies, we need to multiply each percentage by 500. So with that in mind, let's create a table to organize all of our data. So when we do that, we will be able to produce the following table. And the table as we could see, shows our first column, which is the day of the week, and our second column in the middle is our claimed percentage. And our last column is the expected value E, and as you can see, we take our percentage from each day and we write it as a decimal and we multiply it by 500. So going from Sunday to Saturday, so Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday in order, we will find that the expected value for the frequencies will be 50, 90, 80, 70, 60, 50, and 100. Awesome. So now, Moving right along here. Our third step now is we need to perform the T squared calculation. So as we should recall, we should recall that T2. Is equal to. The sum Of parentheses minus E to the power of 2 divided by E. Fantastic. So once again we're calling and using our T squared calculation, we could create another table, and in this table we will produce the following results. So we have our first column, which is the day of the week, then we have our column for O, then we have our column for E. And then we have our column for O minus E to the power 2 divided by E. So going in order for column for O will have 55, 80, 76, 64, 58, 40, 127, and for E, we will have 50, 90, 80, 70, 60, 50, and 100. And for our final column we'll have 0.5, 1.11, 0.20, 0.51, 0.07, 2.00, and 7.29. So when we add these values up, we will find that T2 is equal to 0.5 plus 1.11 plus 0.20 plus 0.51 plus 0.07 plus 2.00 plus 7.29, which will mean that G squared is equal to 11.68. So now, Our fourth step is we need to determine the degrees of freedom and the critical value. So the degrees of freedom, which we'll call this DF and the degrees of freedom, as we should recall, is equal to, for this case, 7 minus 1, which is which, as we should note, is equal to 6. And at our significance level of 0.05, that will mean that our critical value, which you'll call CV for short, our critical value will be equal to 12.592. Fantastic. So now our fifth and final step is we need to make our final decision. So since we note that G2d is equal to 11.68 and this value is less than 12.592, we failed to reject the null hypothesis, so I'll put a big red X, so we failed to reject the null hypothesis. So that will mean When we look at our multiple choice answers, once again, that will mean that our final answer. Without a doubt based on all the information that we determined together, our final answer will have to be the letter B, which once again is there is insufficient evidence to conclude the distribution is different, the claim is reasonable, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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