1. Bell Work 89 The following is a list of test scores from Mrs. Howard’s second period math class: 82, 83, 85, 87, 87, 87, 89, 90, 91, 95, 97, 97. 1. Find the mean, rounded to the nearest whole number. 2. Draw a box-and-whisker plot for the data.

2. Hypothesis testing is used to determine whether the difference in two groups is likely to be caused by chance. Suppose you flipped a coin 20 times. Even if the coin were fair, you would not necessarily get exactly 10 heads and 10 tails. But what if you got 15 heads and 5 tails, or 20 heads and no tails? You might start to think that the coin was not a fair coin, after all.

3. Hypothesis testing cannot be used to prove that a treatment is effective, but it can determine whether the difference between the control group and the treatment group is likely to be caused by chance. In a controlled experiment, there are two groups, the test (or experimental) group and the control group. The hypothesis is the desired or predicted effect of a variable on the test group. By analyzing the minimum, maximum, median, and quartiles of each group, one can prove or disprove the hypothesis.

4. Hypothesis testing begins with an assumption called the null hypothesis. The null hypothesis states that there is no difference between the two groups being tested. The purpose of hypothesis testing is to use experimental data to test the viability of the null hypothesis.

5. Example 1 : Analyzing a Controlled Experiment A. first, state the null hypothesis for the experiment. There is no difference in the scores between the control group and the test group. For this case, let the hypothesis be that scores in the test group are higher than scores in the control group.

6. Example 1: Continued B. Compare the results between the two groups. Start by arranging the data in order to find the median, quartiles, minimum, and maximum. For each data measure, the test group measurement is much higher than the control group. It is highly unlikely the difference is caused by chance; therefore, the null hypothesis should be rejected. Min: 5, Quart 1: 6, Med: 9, Quart 3: 12, Max: 15 Min: 10, Quart 1: 18, Med: 19, Quart 3: 25, Max: 32

7. Example 2 : Analyzing a Controlled Experiment A. State the null hypothesis for the experiment. The blood levels of the drug will be the same for the control group and the treatment group. A medical researcher is testing a new gel coating for a pill, and wants to know if it affects absorption. In a random trial, blood samples were taken from 12 patients in each group 30 minutes after ingesting the pill. The drug levels in micrograms per milliliter are shown below.

8. Example 2: Continued B. Compare the results between the two groups. Start by arranging the data in order to find the median, quartiles, minimum, and maximum. There is a large difference in the two groups that is unlikely to be caused by chance. The researcher should reject the null hypothesis, which could mean that the coating probably does affect absorption. Min: 34, Quart 1: 39, Med: 42, Quart 3: 44, Max: 50 Min: 24, Quart 1: 28, Med: 33, Quart 3: 35, Max: 39

9. Example 2: Continued C. Use box-and-whisker plots to compare the results for the control group (top) and treatment group (bottom)?

10. Example 3 : Analyzing a Controlled Experiment A. State the null hypothesis for the experiment. The glucose levels of the drug will be the same for the control group (A) and the treatment group (B). A researcher is testing whether a certain medication for raising glucose levels is more effective at higher doses. In a random trial, fasting glucose levels of 5 patients being treated at a normal dose (Group A) and 5 patients being treated at a high dose (Group B) were recorded. The glucose levels in mmol/L are shown below.

11. Example 3: Continued B. Compare the results for the control group and the treatment group. Do you think that the researcher has enough evidence to reject the null hypothesis?

12. Example 3: Continued 4.05.06.0 There is a small difference in the two groups that is likely to be caused by chance. If anything, the treatment group actually shows a tendency toward higher glucose levels. The researcher cannot reject the null hypothesis, which means that the medication is probably just as effective at the normal dose as it is at the high dose.

13. Sample Mean and Population Mean The mean is the sum of the values in the set divided by the number of values. It is often represented as x. This is called the Sample Mean The Population Mean is represented by the Greek symbol mu, μ.

14. Z-Test Hypothesis testing can be used to compare the mean from a sample to the mean of a population. If the sample contains at least 30 individuals, you can use the z-test. Suppose that the population mean is estimated to be μ, and a random sample has n individuals (n ≥ 30). To find the z- value of a statistic, you need to know the sample mean x and the standard deviation σ. The z-value is found using the following formula. If │ z │ > 1.96, then you can reject the null hypothesis with 95% certainty. If │ z │ < 1.96, then you do not have enough evidence to reject the null hypothesis.

15. Example 4 : Using a Z-Test The z–value is –4.2, and | z | > 1.96. So, there is enough evidence to reject the null hypothesis. You can say with 95% confidence that the company’s claim about private tutoring is false. A test prep company claims that its private tutoring can boost scores to an average of 2000. In a random sample of 49 students who were privately tutored, the average was 1910, with a standard deviation of 150. Is there enough evidence to reject the claim?

16. Example 5 A tax preparer claims an average refund of $3000. In a random sample of 40 clients, the average refund was $2600, and the standard deviation was $300. Is there enough evidence to reject his claim? Calculate the Z- value: 2600-3000 300 √40 ≈ -400 47.43 ≈-8.43 The z–value is –8.43, and | z | > 1.96. So, there is enough evidence to reject the claim of the tax preparer.

18. Lesson Quiz: Part I 1. A software company is testing whether a new interface decreases the time it takes to complete a certain task. In a random trial, Group A used the existing interface and Group B used the new one. The times in seconds are given for the members of each group. The task will take the same amt. of time for both groups. Group A: 12, 16, 12, 15, 17, 9, 13, 14, 16, 14 Group B: 8, 12, 10, 14, 9, 10, 13, 13, 10, 14 State the null hypothesis for the experiment.

19. Lesson Quiz: Part II Compare the results for Group A and Group B. Do you think that there is enough evidence to reject the null hypothesis? The median of Group B is below the first quartile of Group A. The company can probably reject the null hypothesis. 01020

20. Lesson Quiz: Part III 2. To disprove a previous study that claims that college graduates make an average salary of $46,000, a researcher records the salaries of 50 graduates and finds that the sample mean is $43,000, with a standard deviation of $4,500. What is the z-value, and can she reject the null hypothesis? –4.71; yes