Testing Hypotheses
In Exercises 13–24, assume that a sim...

Question

Testing Hypotheses

In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Got a Minute? Students of the author estimated the length of one minute without reference to a watch or clock, and the times (seconds) are listed below. Use a 0.05 significance level to test the claim that these times are from a population with a mean equal to 60 seconds. Does it appear that students are reasonably good at estimating one minute?

69 81 39 65 42 21 60 63 66 48 64 70 96 91 95

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says a fitness instructor asks a simple random sample of 12 participants to estimate the time it takes to complete a 32nd plank without a timer. Their estimates in seconds are 30, 26, 2021, 29, 27, 26, 35, 35, 26, 30, and 32. At the 0.05 significance level, test the claim that these recorded times are from a population with a mean equal to 30 seconds. Assume the data is normally distributed. So, the first thing we're going to do is actually state our hypotheses. So, for our hypothesis, H knot, We're gonna have that the population mean, mu is going to be equal to 30. And for our alternative hypothesis, H1, we're gonna have that mu is not equal to 30. Next, we need to calculate our sample mean as well as our sample standard deviation because we'll need those to calculate our test statistic. So the first thing we're going to do is calculate our sample mean, which is going to be X bar, and recall that this is going to be equal to the sum of X of I divided by N, where X of I is a data point in our sample, and N is the total number of points in our sample, which in this case is equal to 12. So using the formula X bar is going to be equal to the quantity of 30. Plus 26. Plus 20 Plus 21. Plus 29. Plus 27. Plus 26. Plus 35 plus 35. Plus 26 Plus 30 + 32 in quantity, and all of that is divided by the number of sample points, which is 12. So evaluate this expression, this is going to be equal to. 337 divided by 12, which is approximately equal to 28.083. So next we need to calculate our sample standard deviation, which is S, and recall that that is going to be equal to. The square root of the quantity of the sum. Of the quantity. Of X of I minus X bar in quantity squared divided by the quantity of N minus 1. So, using our data points. That means that our sample standard deviation is going to be equal to. The quantity of 1, divided by the quantity of 12 minus 1 in quantity, multiplied by the quantity. Of for the first term we'll have the quantity of 30 minus 337 divided by 12 in quantity where. Then we'll have plus the quantity of 26 minus 337 divided by 12. In quantity where. They'll have plus. The quantity of 20 minus. 337 divided by 12 in quantity squared. Plus The quantity of 21 minus 337 divided by 12 in quantity squared. Plus The quantity of 29. Minus 337 divided by 12 in quantity squared. Plus, the quantity of 27 minus 337 divided by 12 in quantity squared. Then we'll have plus. The quantity Of 26. Minus 337 divided by 12 in quantity squared. Plus, The quantity of 35 minus 337 divided by 12 in quantity squared. Plus that same quantity again, so we'll have the quantity of 35. Minus 337 divided by 12 in quantity squared. Plus, the quantity of 26 minus 337 divided by 12 in quantity squared. Plus, the quantity of 30. Minus 337 divided by 12, and that quantity is going to be squared. And then finally we'll have less. The quantity of 32 minus 337 divided by 12 in quantity squared. In quantity, then another in quantity raised to the 12 power. So here we just rewritten the square root as raising our quantity to the 1/2 hour. So evaluating this expression, this is going to be approximately equal to. 4. 757. So now we need to calculate our test statistic T, and recall that T. is going to be equal to. The quantity of X bar. Minus U in quantity, divided by the quantity of S divided by the square root of N in quantity. And we actually have values for all those quantities. That means that T is going to be equal to. For X bar, we had 28.083. Minus per mu, we're testing a value of 30. And then that quantity is going to be divided by the quantity of S, which is our standard deviation that we just calculated, which is 4.757. And that is divided by the square root of 12, since N is equal to 12. So, evaluating this quantity, that means that T is going to be approximately equal to. -1.3. 96. So next we need to calculate the number of degrees of freedom, since we'll need those in order to figure out our critical value. So, recall that our degrees of freedom, DF is going to be equal to N minus 1. In this case, N is 12, so this is going to be equal to 12 minus 1. Which is equal to 11. And since that we're performing a two-tail test, that means that our alpha is going to be equal to 0.05, since that is the significance level that we were given in the problem. So, using our table with alpha equal to 0.05, and the degrees of freedom equal to 11, we see that the critical. is equal to 2.201. So, since we have a two-tail test, that means that our critical region is going to be less than minus 2.201. And greater than 2.201. So if we compare RT value to our critical values here, we see that RT value falls in between the two critical values, which means it is outside of the critical area. That means that we fail. To reject the null hypothesis. So, we failed to reject. Each night And so what this means is that At a significant level of 0.05, there is insufficient evidence to warrant rejection of the claim that the recorded times are from a population with a mean equal to 30 seconds. 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.

Key Concepts

Hypothesis Testing

P-value

Significance Level

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