1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 10.3 Hypothesis Testing for Population Means (  Unknown) And some added stuff by D.R.S., University of Cordele. When the population standard deviation is unknown, the sample taken is a simple random sample, and either the sample size is at least 30 or the population distribution is approximately normal, use the t distribution (instead of z) for your hypothesis test.

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The “Test Statistic” in a t test The sample mean The presumed mean, according to the null hypothesis. The sample’s standard deviation The sample size Since a “t” distribution is being used, we’re interested in degrees of freedom, too: d.f. = n - 1 α = the level of significance The Critical Value, t α, which separates out an area of α in the tail(s).

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Left-Tailed Hypothesis Test: H a < μ The critical t value is the negative z value which separates the left tail of area α. tαtα

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Right-Tailed Hypothesis Test: H a > μ The critical t value is the positive z value which separates the right tail of area α. tαtα

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Two-Tailed Hypothesis Test: H a ≠ μ The critical t values are the +/- t values which separate the two tails, area α/2 each. t α/2 -t α/2

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Or use the p-value method 1.Compute test statistic t. 2.What is p-value of that t? 3.Is p-value < the α (alpha) level of significance? as opposed to the Critical value method (below) in which 1. α implies a critical value t α or t α/2. 2. Compute your test statistic t. 3. Is your t more extreme than t α or t α/2 ?

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.19: Finding the Critical t-Value for a Right-Tailed Hypothesis Test Find the critical t-score that corresponds with 14 degrees of freedom at the 0.025 level of significance for a right-tailed hypothesis test. Solution Scanning through the table, we see that the row for 14 degrees of freedom and column for a one-tailed area of  = 0.025 intersect at a critical t-score of 2.145. Hence, TI-84: invT(area to left, degrees of freedom) Here: invT( ______, ______) = ____________ and you have to fix up the __________.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.19: Finding the Critical t-Value for a Right-Tailed Hypothesis Test (cont.) Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails df0.2000.1000.0500.0200.010 131.3501.7712.1602.6503.012 141.3451.7612.1452.6242.977 151.3411.7532.1312.6022.947 161.3371.7462.1202.5832.921 171.3331.7402.1102.5672.898 181.3301.7342.1012.5522.878

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) Nurses in a large teaching hospital have complained for many years that they are overworked and understaffed. The consensus among the nursing staff is that the mean number of patients per nurse each shift is at least 8.0. The hospital administrators claim that the mean is lower than 8.0. In order to prove their point to the nursing staff, the administrators gather information from a simple random sample of 19 nurses’ shifts. The sample mean is 7.5 patients per nurse with a standard deviation of 1.1 patients per nurse.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) (cont.) Test the administrators’ claim using  = 0.025, and assume that the number of patients per nurse has a normal distribution. Solution Step 1:State the null and alternative hypotheses. In order to determine the hypotheses, we must first ask “What do the researchers want to gather evidence for?” H 0 : _______________ H a : ________________

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) (cont.) Step 2:Determine which distribution to use for the test statistic, and state the level of significance. The ___-test statistic is appropriate to use in this case because the claim is about a population ________, the population is normally distributed, the population standard deviation is ____known, and the sample is a simple random sample. In addition to determining which distribution to use for the test statistic, we need to state the level of significance. The problem states that  = ________

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) (cont.)

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) (cont.) You should make a sketch of a bell-shaped curve like this and indicate the critical value, shade the critical region, and indicate where the test statistic lands. Label the t-axis with the critical value and the test statistic value.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.20: Using a Rejection Region in a Hypothesis Test for a Population Mean (Left ‑ Tailed,  Unknown) (cont.) CONCLUSION: { Reject / Fail to Reject } the null hypothesis. INTERPRETATION: We can interpret this conclusion to mean that the evidence collected { is, is not } strong enough at the 0.025 level of significance to reject the null hypothesis in favor of the administrators’ claim that the mean number of patients per nurse is less than 8.0. IN OTHER WORDS – who wins? ____________

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. p-Values Conclusions Using p-Values If p-value ≤ , then reject the null hypothesis. If p-value > , then fail to reject the null hypothesis. Just like with z test, you have two ways of arriving at the same conclusion: The Critical Value Method α determines a rejection region and a critical t value; compare your t test statistic to the critical value – “Is it more extreme? Does it fall in the rejection region?” The p-value method Find the p-value of your test statistic: the area that’s in the tail(s) beyond your p-value. “Is the p-value less than the level of significance, α?”

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) A locally owned, independent department store has chosen its marketing strategies for many years under the assumption that the mean amount spent by each shopper in the store is no more than $100.00. A newly hired store manager claims that the current mean is higher and wants to change the marketing scheme accordingly. A group of 27 shoppers is chosen at random and found to have spent a mean of $104.93 with a standard deviation of $9.07. Assume that the population distribution of amounts spent is approximately normal, and test the store manager’s claim at the 0.05 level of significance.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) (cont.) Step 1:Hypotheses. H 0 : ________ H a :___________ Step 2:Determine which distribution to use for the test statistic, and state the level of significance. A ______-distribution is appropriate to use in this case because the claim is about a population ______, the population is normally distributed, the population standard deviation is ____known, and the sample is a simple random sample. The level of significance is stated in the problem to be  = _______.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) (cont.) Step 3:Gather data and calculate the necessary sample statistics. STAT, TESTS, 2:T-Test the mean, according to the null hypothesis the sample mean the sample standard deviation the sample size what is the alternative hypothesis claiming? Data if raw data in list; Stats if summary here Highlight “Calculate” and press ENTER. Inputs: μ 0 : x: Sx: n: μ: ≠μ 0 μ 0

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) (cont.) Step 3:Gather data and calculate the necessary sample statistics. STAT, TESTS, 2:T-Test Reminder of what the alternative hypothesis says. The t test statistic: The p-value of your t test statistic. Reminder of the sample data you provided.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) (cont.) Step 4:Draw a conclusion and interpret the decision. Compare your p value, _______, to α, _______. Is p < α ? { Yes, No } So the decision is to { Reject, Fail to Reject } the null hypothesis. INTERPRETATION:

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.22: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed,  Unknown) The meat-packing department of a large grocery store packs ground beef in two-pound portions. Supervisors are concerned that their machine is no longer packaging the beef to the required specifications. To test the claim that the mean weight of ground beef portions is not 2.00 pounds, the supervisors calculate the mean weight for a simple random sample of 20 packages of beef. The sample mean is 2.10 pounds with a standard deviation of 0.33 pounds. Is there sufficient evidence at the 0.01 level of significance to show that the machine is not working properly? Assume that the weights are normally distributed.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.22: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed,  Unknown) (cont.) Step 1:hypotheses. H 0 : _______________ H a : ________________ Step 2: Why do we use t in this problem? Because: And for this problem, let’s choose the level of significance ____ = 0.01.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.22: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed,  Unknown) (cont.) Step 3: Gather data and compute. STAT, TESTS, 2:T-Test Inputs: Outputs: Inputs: μ 0 : x: Sx: n: μ: ≠μ 0 μ 0

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.21: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Right-Tailed,  Unknown) (cont.) Step 4:Draw a conclusion and interpret the decision. Compare your p value, _______, to α, _______. Is p < α ? { Yes, No } So the decision is to { Reject, Fail to Reject } the null hypothesis. INTERPRETATION:

25. Excel’s T.TEST is different It seems to deal with two samples. Maybe it’s a tool for Chapter 9. Long story. Disregard it for purposes of this course.