You test a new drug to reduce blood pressure. A group of 15 patients with high blood pressure report the following systolic pressures (measured in mm Hg): before medication: after medication: change: ̄y s 187 120 151 143 160 168 181 197 133 128 130 195 130 147 193 187 118 147 145 158 166 177 196 134 124 133 196 130 146 189 156.40 0 2 4 -2 2 2 4 1 -1 4 -3 -1 0 1 4 1.133 27.409 27.060 2.295 a) Calculate a 90% CI for the change in blood pressure. b) Calculate a 99% CI for the change in blood pressure. c) Does either interval (the one you calculated in (a) or (b)) include 0? Why is this important? d) Now conduct a one sample t-test using μ = 0, and α = .10. Are the results consistent with (a)? e) Finally, conduct a one sample t-test using μ = 0, and α = .01. Are the results consistent with (b)?

What distribution is used for ANOVA? What are limitations of ANOVA?

How do you calculate various portions of the ANOVA process (ex. SST, SSB, SSW)? What do they measure?

What is MSB? MSW? What is the F ratio equal to? What is a computational shortcut for anova (consider how you can calculate SSW)?

How do you calculate degrees of freedom in anova?

Baseball's World Series is a maximum of seven games, with the winner being the first team to win four games. Assume that the Atlanta Braves are playing the Minnesota Twins in the World Series and that the first two games are to be played in Atlanta, the next three games at the Twins' ballpark, and the last two games, if necessary, back in Atlanta. Taking into account the projected starting pitchers for each game and the home field advantage, the probabilities of Atlanta winning each game are as follows:

a. Set up a spreadsheet simulation model in whether Atlanta wins each game is a random variable. What is the probability that the Atlanta Braves win the World Series? If required, round your answer to two decimal places.

b. What is the average number of games played regardless of winner? If required, round your answer to one decimal place.

Suppose that M&M claims that each bag of Peanut M&Ms should be 18 grams and Plain M&Ms should be 13.5 grams.

a. Test the claim that M&M is shorting its customers in bags of Plain M&Ms.

b. Test the claim that M&M is overfilling Peanut bag bags of M&Ms.

c. Discuss your choice of ?.

i. Why did you choose the ? you did?

ii. If you had chosen a different ?, would it have affected your conclusion?

Total Plain M&Ms: 665

Total Peanut M&Ms: 356

Sum Total Weight of All the Bags of Peanut M&Ms: 921 grams

(18g, 21g, 19g, 17g, 18g, 21g, 18g, 18g, 21g, 17g, 19g, 16g, 20g, 20g, 17g, 17g, 18g, 16g, 20g, 18g, 19g, 20g, 20g, 18g, 21g, 19g, 17g, 18g, 17g, 19g, 16g, 19g, 19g, 19g, 17g, 20g, 18g, 18g, 17g, 19g, 19g, 18g, 18g, 18g, 17g, 17g, 19g, 20g, 18g, 18g)

Sum Total Weight of All the Bags of Plain M&Ms: 601 grams

(13g, 13g, 12g, 14g, 14g, 14g, 13g, 13g, 12g, 12g, 13g, 13g, 13g, 14g, 13g, 14g, 12g, 11g, 12g, 12g, 13g, 10g, 15g, 14g, 16g, 14g, 15g, 13g, 12g, 13g, 14g, 13g, 13g, 13g, 11g, 12g, 12g, 14g, 13g, 13g, 14g, 14g, 12g, 13g, 13g, 15g)

The following data represent the calories and​ sugar, in​ grams, of various breakfast cereals.

Use the data above to complete parts​ (a) through​ (d).

Compute the covariance.

b. Compute the coefficient of correlation.

c. Which do you think is more valuable in expressing the relationship between calories and

sugarlong dash—the

covariance or the coefficient of correlation? Explain.

d. What conclusions can you reach about the relationship between calories and sugar?

1) You want to construct a 92% confidence interval. The correct z* to use is

A) 1.645

B) 1.41

C) 1.75

2) Suppose the average Math SAT score for all students taking the exam this year is 480 with standard deviation 100. Assume the distribution of scores is normal. The senator of a particular state notices that the mean score for students in his state who took the Math SAT is 500. His state recently adopted a new mathematics curriculum, and he wonders if the improved scores are evidence that the new curriculum has been successful. Since over 10,000 students in his state took the Math SAT, he can show that the P-value for testing whether the mean score in his state is more than the national average of 480 is less than 0.0001. We may correctly conclude that

A) these results are not good evidence that the new curriculum has improved Math SAT scores.

B) there is strong statistical evidence that the new curriculum has improved Math SAT scores in his state.

C) although the results are statistically significant, they are not practically significant, since an increase of 20 points is fairly small.

3) Suppose the average Math SAT score for all students taking the exam this year is 480 with standard deviation 100. Assume the distribution of scores is normal. A SRS of four students is selected and given special training to prepare for the Math SAT. The mean Math SAT score of these students is found to be 560, 80 points higher than the national average. We may correctly conclude

A) the results are statistically significant at level α = 0.05, but they are not practically significant.

B) the results are not statistically significant at level α = 0.05, but they are practically significant.

C) the results are neither statistically significant at level α = 0.05 nor practically significant.

Systematic random samples are often used to choose a sample of apartments in a large building or dwelling units in a block at the last stage of a multistage sample. An example will illustrate the idea of a systematic sample. Suppose that we must choose 5 addresses out of 125. Because 125/5=25, we can think of the list as five lists of 25 addresses. Choose 1 of the first 25 at random, using software or Table B. The sample contains this address and the addresses 25, 50, 75, and 100 places down the list from it. If 13 is chosen, for example, then the systematic random sample consists of the addresses numbered 13, 38, 63, 88, and 113.

(a) A study of dating among college students wanted a sample of 200 of the 8000 single male students on campus. The sample consisted of every 40^th name from a list of the 8000 students. Explain why the survey chooses every 40^th name.

(b) Use software or Table B at line 112 to choose the starting point for this systematic sample.

I have a ratio that in total doesn't equal to 100%, how would you explain the reason for that? The data is of 13 people. I have data Male 8 people= 61.54%, Female 5 people = 38.36%;//// Unemployed 6 people=46.15%, Food industry 3 people 23.08%;//// British 2 people=15.38%, Chinese 5 people 38.46%, Hispanic 3 people=23.08%////Agnostic 5 people=38.46%, Christian 5 people=38.46%. When you add each section those equal to 100%. But when I have England 2 people=15.38%, USA 2 people=15.38%, China 5 people 38.46% I'm off by 0.01%. 15.38+15.38+38.46=69.22; so 100-69.22=30.78. The 4 people out of 13 people would equal 30.77. So I'm off 0.01%.

Given: "The proportion of left-handed people is equal to 10%"

a) Provide a Conclusion that would be a Type I error

b) Provide a Conclusion that would be a Type II error

Tomato weights and Fertilizer (Raw Data, Software Required):
Carl the farmer has three fields of tomatoes, on one he used no fertilizer, in another he used organic fertilizer, and the third he used a chemical fertilizer. He wants to see if there is a difference in the mean weights of tomatoes from the different fields. The sample data for tomato-weights in grams is given below. Carl claims there is a difference in the mean weight for all tomatoes between the different fertilizing methods.

The Test: Complete the steps in testing the claim that there is a difference in the mean weight for all tomatoes between the different fertilizing methods.

(a) What is the null hypothesis for this test?

H₀: At least one of the population means is different from the others.

H₀:  μ₁ ≠ μ₂ ≠ μ₃.    

H₀:  μ₁ = μ₂ = μ₃.

H₀:  μ₃ > μ₂ > μ₁.

(b) What is the alternate hypothesis for this test?

H₁:  μ₁ ≠ μ₂ ≠ μ₃.

H₁:  μ₁ = μ₂ = μ₃.   

H₁: At least one of the population means is different from the others.

H₁:  μ₃ > μ₂ > μ₁.

(c) Use software to get the P-value of the test statistic ( F ). Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =  

(d) What is the conclusion regarding the null hypothesis at the 0.10 significance level?

reject H₀

fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that all of the mean weights are the same.

There is sufficient evidence to conclude that the mean weights are different.    

There is not enough evidence to conclude that the mean weights are different.

(f) Does your conclusion change at the 0.05 significance level?

Yes

No    

Tomato weights and Fertilizer: Carl the farmer has three fields of tomatoes, on one he used no fertilizer, in another he used organic fertilizer, and the third he used a chemical fertilizer. He wants to see if there is a difference in the mean weights of tomatoes from the different fields. The sample data is given below. The second table gives the results from an ANOVA test. Carl claims there is a difference in the mean weight for all tomatoes between the different fertilizing methods.

Tomato-Weight in Grams

ANOVA Results

The Test: Complete the steps in testing the claim that there is a difference in the mean weight for all tomatoes between the different fertilizing methods.

(a) What is the null hypothesis for this test?

H₀:  μ₁ ≠ μ₂ ≠ μ₃.

H₀: At least one of the population means is different from the others.    

H₀:  μ₃ > μ₂ > μ₁.

H₀:  μ₁ = μ₂ = μ₃.

(b) What is the alternate hypothesis for this test?

H₁:  μ₁ ≠ μ₂ ≠ μ₃.

H₁: At least one of the population means is different from the others.    

H₁:  μ₃ > μ₂ > μ₁

.H₁:  μ₁ = μ₂ = μ₃.

(c) What is the conclusion regarding the null hypothesis at the 0.01 significance level?

reject H₀

fail to reject H₀    

(d) Choose the appropriate concluding statement.

We have proven that all of the mean weights are the same.

There is sufficient evidence to conclude that the mean weights are different.    

There is not enough evidence to conclude that the mean weights are different.

(e) Does your conclusion change at the 0.10 significance level?

Yes

No    

Demonstrating that a correlation exists does not prove that changes in one variable are the cause of changes in the other, partly because other factors which are undetected may be influencing both known variables. Thus, knowing that a correlation exists may lead to two or more different interpretations of the correlation. For each of the studies described below, decide whether the correlation is positive or negative and give two explanations for the finding.

1 A government study reveals that the more a mother smokes, the more likely her children are to exhibit behavioral problems.

2 The more psychology courses students take during their college years, the higher their scores on a measure of interpersonal sensitivity.

3 A survey of adolescents being treated for eating disorders noted that those who watched the most TV during the week tended to receive the lowest ratings on a measure of general health.

Wait-Times (Raw Data, Software Required):
There are three registers at the local grocery store. I suspect the mean wait-times for the registers are different. The sample data is depicted below. It gives the wait-times in minutes.

The Test: Complete the steps in testing the claim that there is a difference in mean wait-times between the registers.

(a) What is the null hypothesis for this test?

H₀:  μ₂ > μ₃ > μ₁.

H₀: At least one of the population means is different from the others.    

H₀:  μ₁ = μ₂ = μ₃.

H₀:  μ₁ ≠ μ₂ ≠ μ₃.

(b) What is the alternate hypothesis for this test?

H₁:  μ₁ = μ₂ = μ₃.

H₁: At least one of the population means is different from the others.    

H₁:  μ₁ ≠ μ₂ ≠ μ₃.

H₁:  μ₂ > μ₃ > μ₁.

(c) Use software to get the P-value of the test statistic ( F ). Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =  

(d) What is the conclusion regarding the null hypothesis at the 0.05 significance level?

reject H₀

fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that all of the mean wait-times are the same.There is sufficient evidence to conclude that the mean wait-times are different.    There is not enough evidence to conclude that the mean wait-times are different.

(f) Does your conclusion change at the 0.01 significance level?

Yes

No    

The judges of County X try thousands of cases per year. Although in a big majority of the cases disposed the verdict stands as rendered, some cases are appealed. Of those appealed, some are reversed. Because appeals are often made as a result of mistakes by the judges, you want to determine which judges are doing a good job and which ones are making too many mistakes. The attached Excel file has the results of 182,908 disposed cases over a three year period by the 38 judges in various courts of County X. Two of the judges (Judge 3 and Judge 4) did not serve in the same court for the entire three-year period. Using your knowledge of probability and conditional probability you will make an analysis to decide a ranking of judges. You will also analyze the likelihood of appeal and reversal for cases handled by different courts. Using Excel to calculate the following probabilities for each judge. Use the attached Excel file to fill in these numbers on the tables for each court. Each court is given on a separate tab. The probability of a case being appealed for each judge. The probability of a case being reversed for each judge. The probability of a reversal given an appeal for each judge. Probability of cases being appealed and reversed in the three courts. Rank the judges within each court.

Answer the questions and interpret the data.

Colleen is the marketing manager for Virtually Viral, an entertainment company that collects viral videos from around the Internet and aggregates them on their website. Whether it’s videos of cats or unusual marriage proposals, Virtually Viral collects them all. Almost all of Virtually Viral’s revenue comes from clicks on advertisements surrounding the videos. To maximize profits, Colleen tries to match ad content to video content. For example, for the ‘Wacky Weddings’ section of the website, most advertisements link to wedding planners and invitation/paper product suppliers. As part of this effort, Colleen contracted a web design firm to put together a new look for the website, with the goal of improving the amount of time visitors spend on the website. They produced four different versions, each arranging the videos and advertisements differently. Colleen is unsure which of these designs would result in the greatest amount of time spent on the site. To solve this problem, Colleen designs an experiment. She sets up a system to randomly assign visitors to the website to experience one of the four designs, recording the number of seconds that they spend on the site. She wants to compare the groups with each other and see if the different designs result in different lengths of time viewing the website. Whichever results in the longest visits will become the new design for the site in general. She knows from Chapter 7 that she has a research question and that this calls for some type of hypothesis testing. In Chapter 9, she learned that treating groups differently and comparing them means that she has independent data. But the independent-samples t-test only compares two groups with each other and she has four. Should she run multiple independent-samples t-tests? Or is there a better way?

Colleens problem can be solved with an ANOVA and post-hoc test.  

DATA